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Lightning surge propagation in overhead lines and gas insulated bus-ducts and cables Lee, Kai-Chung 1980

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LIGHTNING SURGE PROPAGATION IN OVERHEAD LINES AND GAS INSULATED BUS-DUCTS AND CABLES  by  |^LEE  KAI-CHUNG  B.ScT, U n i v e r s i t y o f W i s c o n s i n , 1973 M.Sc., U n i v e r s i t y o f B r i t i s h Columbia, 1975 M.A.Sc., U n i v e r s i t y o f B r i t i s h Columbia, 1977  r A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  The Faculty o f Graduate Studies i n the Department '_ of Electrical  We accept t h i s to the r e q u i r e d  THE  Engineering  t h e s i s as conforming standard  UNIVERSITY OF BRITISH COLUMBIA J u l y , 1980  0  Lee Kai-Chung, 1980  In p r e s e n t i n g t h i s  thesis  an advanced degree at  further  fulfilment  of  the  requirements  the U n i v e r s i t y of B r i t i s h Columbia, I agree  the L i b r a r y s h a l l make it I  in p a r t i a l  freely  available  for  this  thesis  f o r s c h o l a r l y purposes may be granted by the Head of my Department  of  this thesis for  It  financial  i s understood that copying or gain s h a l l not  written permission.  Department  of  ____________________  The U n i v e r s i t y of B r i t i s h Columbia 2075  Wesbrook  Vancouver, V6T  Date  1WS  Place  Canada  that  reference and study.  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f  by h i s r e p r e s e n t a t i v e s .  for  or  publication  be allowed without my  ABSTRACT  The p r o p a g a t i o n c h a r a c t e r i s t i c s of l i g h t n i n g surges i n compressed SFg gas  i n s u l a t e d power s u b s t a t i o n was  t r a n s i e n t s program.  studied  u s i n g an  Numerical models were developed  b e h a v i o u r of d i f f e r e n t  electromagnetic  t o r e p r e s e n t the  system components e s p e c i a l l y under l i g h t n i n g  over-  voltage conditions. The  c h a r a c t e r i s t i c s of l i g h t n i n g surge p r o p a g a t i o n i n  m u l t i - p h a s e untransposed  t r a n s m i s s i o n l i n e s was  analysed  overhead first.  Modal  a n a l y s i s , :tpgether w i t h s p e c i a l r o t a t i o n t e c h n i q u e s t o f i t time domain s o l u t i o n s were then used t o s i m u l a t e the wave p r o p a g a t i o n untransposed  l i n e i n an e l e c t r o m a g n e t i c t r a n s i e n t s program.  voltage-dependent investigated.  i n multi-phase Non-linear  corona a t t e n u a t i o n and d i s t o r t i o n phenomena were a l s o  A v a i l a b l e f i e l d t e s t r e s u l t s c o u l d be d u p l i c a t e d t o w i t h i n  5%. The  c h a r a c t e r i s t i c s of l i g h t n i n g surge p r o p a g a t i o n i n m u l t i - p h a s e  s i n g l e - c o r e SF^ c a b l e s was  studied next.  A program was  the c a b l e parameters f o r t y p i c a l c a b l e c o n f i g u r a t i o n s . c u r r e n t r e t u r n i n g through to  illustrate  i t s own  sheath and through  developed  The amount o f c o r e  the e a r t h were computed  the s i n g l e phase c a b l e r e p r e s e n t a t i o n f o r wave  p r o p a g a t i o n i n s i n g l e core SF  ft  cables.  ii  to o b t a i n  TABLE OF CONTENTS Page ABSTRACT  i i  TABLE OF CONTENTS  i i i  ACKNOWLEDGEMENTS  vi  INTRODUCTION  1  CHAPTER 1 - LIGHTNING CHARACTERISTICS AND STROKES TO POWER TRANSMISSION LINES 1.  Introduction  3  2.  L i g h t n i n g d i s c h a r g e mechanism  3  3.  S t a t i s t i c a l characteristics  of l i g h t n i n g  strokes  7  4.  Frequency o f l i g h t n i n g  s t r o k e s to e a r t h  5.  Frequency o f l i g h t n i n g  s t r o k e s t o power l i n e s  6.  S h i e l d i n g f a i l u r e phenomenon o f l i g h t n i n g strokes  CHAPTER 2 - LIGHTNING SURGE PROPAGATION TRANSMISSION LINES  9 . .  12  13  IN OVERHEAD  1.  Introduction  18  2.  Modal a n a l y s i s f o r N-phase untransposed l i n e . . .  19  3.  R o t a t i o n of e i g e n v e c t o r s conductance  f o r zero  22  C o n f i r m a t i o n o f accuracy eigenvector subroutine  o f e i g e n v a l u e and  4.  5.  6.  7.  8.  Real-valued matrix  shunt  25  frequency-independent  transformation 25  Frequency dependent e f f e c t propagation  in lightning  surge 27  D e t e r m i n a t i o n o f surge impedance o f t h e s t r u c k phase of a t r a n s m i s s i o n l i n e  33  S i n g l e phase r e p r e s e n t a t i o n f o r c l o s e - b y on double c i r c u i t e d l i n e  36  iii  strokes  CHAPTER 3 - LIGHTNING WAVE PROPAGATION IN S F GAS INSULATED UNDERGROUND TRANSMISSION CABLE SYSTEM &  1.  Introduction  46  2.  Formation o f s e r i e s impedance m a t r i x f o r SF, c a b l e s 6  49  3.  C a l c u l a t i o n o f s e l f and mutual e a r t h r e t u r n . . . .  50  4.  C a l c u l a t i o n of s e l f s i n g l e core c a b l e .  61  5.  6.  7.  8.  9.  10.  impedance m a t r i x f o r  Sheath c u r r e n t r e t u r n c h a r a c t e r i s t i c s f o r usual earth  67  Sheath c u r r e n t r e t u r n c h a r a c t e r i s t i c s f o r s u b s t a t i o n e a r t h w i t h grounding g r i d network . . .  74  Formation o f shunt admittance m a t r i x cables  79  f o r SFg  Confirmation of numerical accuracy f o r cable parameter c a l c u l a t i o n and c u r r e n t r e t u r n r a t i o  . .  80  S i n g l e phase r e p r e s e n t a t i o n parameters f o r m u l t i - p h a s e SF^ c a b l e s  80  Wave p r o p a g a t i o n  84  i n SF^ c a b l e s  CHAPTER 4 - CORONA ATTENUATION AND DISTORTION CHARACTERISTICS OF LIGHTNING OVERVOLTAGE IN OVERHEAD TRANSMISSION LINES 1.  Introduction  87  2.  P h y s i c a l p r o p e r t i e s of corona a t t e n u a t i o n and distortion characteristics  87  3.  Transmission  4.  S o l u t i o n o f l i n e equations by compensation method w i t h t r a p e z o i d a l r u l e s  91  I n f l u e n c e on corona by a d j a c e n t i n t h e same bundle  sub-conductors 97  6.  I n f l u e n c e on corona by a d j a c e n t  phase c o n d u c t o r s  7.  Optimal lumping l o c a t i o n s and number o f corona  5.  l i n e equations  f o r coronated  lines.  .  .  branch l e g s 8.  O v e r a l l numerical  90  99  99 modelling iv  f o r corona e f f e c t s  . .  102  CHAPTER 5 - CONCLUSIONS  106  APPENDIX A - SKIN DEPTH ATTENUATION IN CONDUCTING MEDIUM WITH FINITE CONDUCTIVITY  107  BIBLIOGRAPHY  109  v  ACKNOWLEDGEMENT  I would  l i k e t o show my deepest a p p r e c i a t i o n to my  thesis  s u p e r v i s o r P r o f e s s o r Hermann W. Dommel f o r p r o v i d i n g such a unique  chance  i n d o i n g r e s e a r c h i n e l e c t r o m a g n e t i c t r a n s i e n t s o f power systems. Dr. Dommel's v a l u a b l e c r i t i c i s m , and c o u n t l e s s hours o f d i s c u s s i o n d u r i n g r e s e a r c h are a l s o g r a t e f u l l y  acknowledged.  I am g r a t e f u l t o the B.C.  Hydro E n g i n e e r s , Messrs. J a c k Sawada,  Brent Hughes, Ken N i s h i k a w a r a and N i c k Cuk f o r t h e i r h e l p f u l I am a l s o t h a n k f u l t o my  discussions.  f e l l o w graduate s t u d e n t s and c o l l e a g u e s Messrs.  Obed Abledu, But-Chung C h i u , S h i Wei f o r t h e i r d i f f e r e n t p o i n t s of view. The f i n a n c i a l support from the U n i v e r s i t y o f B r i t i s h Columbia i n form of t e a c h i n g and r e s e a r c h a s s i s t a n t s h i p and f e l l o w s h i p i s v e r y much appreciated.  The f i n a n c i a l a s s i s t a n c e o f the Systems E n g i n e e r i n g D i v i s i o n  of the B r i t i s h Columbia Hydro and Power A u t h o r i t y through a Power Systems Research Agreement, and the B r i t i s h Columbia Telephone Company Scholarship are also g r a t e f u l l y  Graduate  acknowledged.  S p e c i a l thanks are expressed t o Miss G a i l Hrehorka i n the E l e c t r i c a l E n g i n e e r i n g Main O f f i c e f o r p r o d u c i n g t h i s e x c e l l e n t l y typed t h e s i s . F i n a l l y , I am  i n d e b t e d to my p a r e n t s f o r t h e i r c o n t i n u o u s  encouragement and my w i f e f o r her p a t i e n c e .  vi  1 INTRODUCTION  Every y e a r , atmospheric  l i g h t n i n g d i s c h a r g e s cause numerous  d i s t u r b a n c e s and damages to e l e c t r i c power systems, such a s , d e s t r o y i n g transformers devoted  and c a u s i n g b l a c k - o u t s of l a r g e a r e a s .  This thesis i s  t o the a n a l y s i s of l i g h t n i n g surge p r o p a g a t i o n i n t o compressed  SFg g a s - i n s u l a t e d s u b s t a t i o n s . Hydro and Power A u t h o r i t y was  The MICA p r o j e c t of the B r i t i s h  chosen as a t e s t  I n s u l a t i o n c o - o r d i n a t i o n requirements s i m u l a t e d surge p r o p a g a t i o n s t u d i e s .  Columbia  example.  are u s u a l l y d e r i v e d from  T h i s t h e s i s shows t h a t the p r e s e n t  p r a c t i c e of i n s u l a t i o n c o - o r d i n a t i o n d e s i g n can be improved w i t h  the  n u m e r i c a l models developed  this  in this thesis.  The c o n t r i b u t i o n s o f  t h e s i s t o i n s u l a t i o n c o - o r d i n a t i o n d e s i g n and r e l a t e d power system s t u d i e s i n c l u d e s the 1.  following:  D e t e r m i n a t i o n of wave p r o p a g a t i o n i n untransposed  lines - Analysis i s  used, w i t h a s p e c i a l r o t a t i o n of modal parameters and t r a n s f o r m a t i o n m a t r i c e s to make the method s u i t a b l e f o r time-domain s o l u t i o n s of wave p r o p a g a t i o n of d i f f e r e n t  i n m u l t i - p h a s e untransposed  simplified  line.  The  suitability  t r a n s m i s s i o n l i n e models i s c l a r i f i e d  comparing s i m u l a t i o n r e s u l t s w i t h those from an exact  by  multi-phase  rep r e s e n t a t i o n . 2.  R e p r e s e n t a t i o n of n o n - l i n e a r voltage-dependent  corona e f f e c t s - Corona  d i s t o r t i o n and a t t e n u a t i o n has been s i m u l a t e d w i t h v o l t a g e dependent 44 v e l o c i t i e s and c o r r e c t i o n f a c t o r s i n the p a s t d i f f e r e n c e methods.  12 '  , or with  finite  However, these methods are i n e f f i c i e n t f o r  d i g i t a l computer a p p l i c a t i o n s .  More e f f i c i e n t  u s i n g compensation methods, a r e developed  computational  i n t h i s t h e s i s to  algorithms,  2 i n v e s t i g a t e the n o n - l i n e a r v o l t a g e dependent corona 3.  Determination  o f wave p r o p a g a t i o n i n m u l t i - p h a s e  effects.  s i n g l e c o r e SF^  c a b l e s - P u b l i s h e d methods f o r t h e c a l c u l a t i o n o f c a b l e c o n s t a n t s give inconsistent r e s u l t s .  A new c a b l e c o n s t a n t s program f o r m u l t i -  phase s i n g l e core SF^-cables has been developed various converging.Mnfinite s e r i e s .  by the a u t h o r , u s i n g  The complete s h i e l d i n g  effect  of t h e e x t e r n a l l y grounded sheath a t f r e q u e n c i e s above 1 k Hz has been confirmed w i t h t h i s program. The problem o f t r a n s i e n t g r o u n d r i s e caused or by l i g h t n i n g  by i n t e r n a l breakdowns  impulses, as s t u d i e d by O n t a r i o Hydro"^, i s not i n c l u d e d  i n t h i s t h e s i s . These t r a n s i e n t p o t e n t i a l d i f f e r e n c e s between SF^ busd u c t s and ground o c c u r m a i n l y a t t h e j u n c t i o n w i t h t h e overhead As t h i s t h e s i s shows, t h e c u r r e n t r e t u r n i n t h e SF  bus-duct  line.  i s completely  o through t h e sheath a t f r e q u e n c i e s above 1 kHz, whereas t h e c u r r e n t r e t u r n of t h e l i g h t n i n g  impulse  on t h e overhead  l i n e i s i n t h e ground. At t h e  j u n c t i o n , t h e r e t u r n c u r r e n t must t h e r e f o r e pass from t h e sheath  into  the ground through t h e ground l e a d s , which i n t u r n causes t h e t r a n s i e n t g r o u n d r i s e problem. These t r a n s i e n t g r o u n d r i s e s a r e an important i n t h e d e s i g n of t h e grounding  factor  system, because they can cause damage t o r u  a u x i l i a r y w i r i n g or shocks t o p e r s o n n e l .  3  CHAPTER 1:  1.  LIGHTNING CHARACTERISTICS AND TO POWER TRANSMISSION LINES  STROKES  Introduction The  first  important  experiment on l i g h t n i n g was done by Benjamin  F r a n k l i n , who used f l y i n g k i t e s t o show t h a t l i g h t n i n g i s e l e c t r i c a l i n nature.  F o r more than two c e n t u r i e s , l i g h t n i n g has been t h e s u b j e c t o f  active research.  Much o f t h i s r e s e a r c h has been concerned w i t h  t e c t i o n o f p e o p l e and p r o p e r t y a g a i n s t t h e e f f e c t s o f l i g h t n i n g  2.  stroke.  L i g h t n i n g d i s c h a r g e mechanisms Lightning strokes are f i r s t  thunder-cloud  u s u a l l y contains  initiated  i n s i d e thunder-clouds.  several negative  electron  i s jumping  over  to  A  and p o s i t i v e charge  d i s t r i b u t e d i n d i f f e r e n t l o c a t i o n s as shown i n F i g u r e 1.1a. the  the p r o -  centres  As soon as  n e u t r a l i z e t h e p o s i t i v e charge,  a step l e a d e r s t a r t s t o move down t h e e a r t h i n d i s c r e t e z i g - z a g s t e p s o f about 50 meters i n l e n g t h as shown i n F i g u r e 1.1b.  T h i s downward  pilot  s t r o k e i s about 1 cm i n diameter and i s not v i s u a l l y d e t e c t a b l e by the human eye. As t h i s stepped l e a d e r c o n t i n u e s charges a r e induced  to progress  downwards, p o s i t i v e  and accumulated on t h e ground s u r f a c e .  these p o s i t i v e charges  jump  Eventually,  upwards and form t h e r e t u r n s t r o k e to  meet the downward stepped l e a d e r as shown i n F i g u r e 1.1c.  This highly  luminous r e t u r n s t r o k e produces most o f the thunder which i s h e a r d .  The  r e t u r n s t r o k e i s about 10 cm i n diameter and a t a temperature o f around 30,000°K.  Once a l l the p o s i t i v e charges t r a n s f e r t o the t h u n d e r - c l o u d  shown i n F i g u r e l . l d ,  the d i s c h a r g e d  charge c e n t r e completely  becomes  as  4  a.  Charge n e u t r a l i z a t i o n w i t h i n the c l o u d .  b.  Stepped l e a d e r moving downwards.  I n i t i a l i z a t i o n o f upward moving r e t u r n l e a d e r .  d.  Complete upward p r o p a g a t i o n of r e t u r n l e a d e r to c l o u d (charge c e n t r e becomes positive).  F i g u r e 1.1:  Charge d i s t r i b u t i o n and p r o p a g a t i o n i n i t i a l l i g h t n i n g discharge.  during  5  a.  c.  D i s c h a r g e between 2 charge c e n t r e s .  N e g a t i v e charge d a r t s t r o k e about to h i t the ground.  F i g u r e 1.2:  b.  d.  N e g a t i v e charge d a r t s t r o k e f l o w i n g down the continuous e a r t h p a t h .  Formation of subsequent r e t u r n l e a d e r from ground t o c l o u d charge c e n t r e s .  Charge d i s t r i b u t i o n and p r o p a g a t i o n d u r i n g subsequent d a r t l e a d e r ' ( m u l t i p l i - s t r o k e l i g h t n i n g ) .  6  p o s i t i v e and  s i n g l e stroke l i g h t n i n g discharge  However, about 50% and  of a l l l i g h t n i n g f l a s h e s are m u l t i - s t r o k e s  c o n t a i n 3 or 4 subsequent s t r o k e s , t y p i c a l l y separated  About l e s s than 100 is  1 i s completed.  ms  a f t e r the f i r s t  by 30  2 3 '  to 40  ms.  stroke, a high p o t e n t i a l difference  a g a i n e s t a b l i s h e d between the charge c e n t r e s .  Discharges a g a i n occur  and  a d a r t l e a d e r i s formed which moves earthwards i n the p r e v i o u s main stream as shown i n F i g u r e 1.2a  to 1.2c.  S i m i l a r l y , a r e t u r n s t r o k e i s a l s o formed  and more p o s i t i v e charges t r a n s f e r Figure  process  of m u l t i - s t r o k e s w i t h r e l a t i v e s t r o k e magnitudes  time s c a l e s i s i l l u s t r a t e d  sequent s t r o k e s a r e (about  as shown i n  1.2d. The whole  and  to the thunder-clouds  30 - 40 ms)  i n F i g u r e 1.3a  of the i n i t i a l i n time. The  T y p i c a l sub-  s t r o k e magnitude and  initial  i n s u l a t i o n c o - o r d i n a t i o n s t u d i e s , but  and- 1.3b.  are w e l l  separated  s t r o k e i s t h e prime f a c t o r i n the  subsequent s t r o k e s must be taken i n t o  account as a r r e s t e r s must be a b l e to handle r e p e t i t i v e d i s c h a r g e s , and dead times of the a u t o - r e c l o s i n g switchgear  V e l o c i t y 100%  F i g u r e 1.3a:  = 300  must be  set  longer.  m/jjs  Diagram showing time i n t e r v a l s between and subsquent s t r o k e s . ( R e f . 3 )  initial  the  100|is  lOQuis -35ms-  F i g u r e 1.3b:  3.  •35ms  C u r r e n t magnitudes of i n i t i a l and subsequent strokes i n t y p i c a l l i g h t n i n g flashes.  Statistical characteristics Due  inside  of l i g h t n i n g  to the d i f f e r e n t d i s t r i b u t i o n s  the t h u n d e r - c l o u d ,  and  the c h a r a c t e r i s t i c s  large s t a t i s t i c a l variation  strokes i n t e n s i t i e s of charge of l i g h t n i n g  i n both magnitude and  a.  Magnitude of l i g h t n i n g  The  voltage stress  s t r o k e s show a  shape.  strokes  on the power system  the magnitude of the l i g h t n i n g  centres  depends  on  c u r r e n t , which i s t h e r e f o r e a  critical 4  factor  i n determining  i n s u l a t i o n requirements.  are shown i n F i g u r e 1.4. I t can be magnitudes are w i t h i n 10 t o 100 kA,  52  -Recorded measurements .'  seen t h a t 80% of the l i g h t n i n g c u r r e n t and o n l y 5% exceed magnitude o f 100 kA.  52 It i s as an i n c i d e n t  suggested  t h a t the l i g h t n i n g  c u r r e n t source  magnitude of 100  kA  due  to be  simulated  to the power l i n e w i t h a maximum c u r r e n t  for insulation co-ordination studies.  c u r r e n t source w i l l become an o v e r v o l t a g e power l i n e  s t r o k e has  to the i n h e r e n t  However, t h i s  wave when p r o p a g a t i n g  surge impedance of the l i n e .  down the Thus, f o r  8  F i g u r e 1.4:  Cumulative P r o b a b i l i t y o f Occurrence of the Amplitudes of L i g h t n i n g C u r r e n t s o b t a i n e d by summarising r e s u l t s from more than 624 measured i n c i d e n t s from 9 c o u n t r i e s (Ref. 4 ) .  9  equipment t e s t purposes, o v e r v o l t a g e  waves are p r e s c r i b e d ,  b.  Waveshape of l i g h t n i n g  stroke  The  l i g h t n i n g waveshapes measured by d i f f e r e n t r e s e a r c h e r s  essen-  t i a l l y resemble a double e x p o n e n t i a l waveshape of d i f f e r e n t r i s e time  and  decay time. The  us.  observed spread  of r i s e  time i s from v e r y s h o r t t o 10 5  The  observed decay time a l s o spreads from 2 to 100  ps  (see F i g u r e  The  e l e c t r i c power i n d u s t r y t h e r e f o r e agreed many y e a r s  ago  1.5a).  to use  a  l i g h t n i n g o v e r v o l t a g e wave f o r equipment i n s u l a t i o n t e s t i n g purpose of a shape 1.2  x 50 ys  ( e x p l a n a t i o n - o f d e s i g n a t i o n i n F i g u r e 1.5b  Some t e s t i n g p r e c r i p t i o n s a l s o s p e c i f y t h a t t h i s f u l l wave be a spark gap  i n the t a i l  which a r e c o n t a i n e d  4.  level.  earth  thunderstorm a c t i v i t y on e a r t h i s measured by the  This isokeraunic l e v e l  r e g i o n s c l o s e to the  IKL  (IKL)  isokeraunic  g i v e s the number of days per year  U s u a l l y , thunder cannot be heard o u t s i d e a  An updated world map  As e x p e c t e d , h i g h e r  frequencies  i n the v o l t a g e c o l l a p s e .  thunder has been h e a r d . radius.  of i s o k e r a u n i c l e v e l  e a r t h per km  2  (N)  i s found w i t h i n the t r o p i c a l and s u b - t r o p i c a l  equator.  i n a p a r t i c u l a r l o c a t i o n i s given  N = A (IKL) s t r o k e / ( k m  where A = 0.1  to  0.2  2  - yr)  by  7  that  7-24  i s shown i n F i g u r e  A f t e r o b t a i n i n g the IKL o f a g i v e n p l a c e , the number o f to  1.5c).  chopped w i t h  to expose the equipment to the h i g h e r  Frequency of l i g h t n i n g s t r o k e s to The  and  strokes  1.6.  km  10  t ^ = r i s e time =time t o c r e s t t2=decay time =time to h a l f v a l u e  10  a.  100  T  Wave f r o n t s and t a i l s o f l i g h t n i n g  i  m  e  ^  s  )  surges  Time  Double e x p o n e n t i a l wave V = v ^ ( e  -*t  e  s wave t  t  1  r  )  Time to c r e s t =1.67(X -X ) . =1.2jus  Time t o h a l f =X.-X I 4 o =50JJS  Time  c.  Typical  F i g u r e 1.5:  surge waveform impulse  Waveshape o f l i g h t n i n g  2  generator  strokes.(Ref.5)  value  F i g u r e 1.6:  World D i s t r i b u t i o n of Thunderstorm Days  (Ref.6)  12  5.  Frequency of l i g h t n i n g  s t r o k e s t o power  F o r e s t i m a t i n g t h e number of l i g h t n i n g we can s t a r t ,  ' e l e c t r i c a l shadow'  power l i n e s . chosen.  s t r o k e s t o power  lines,  from the ' e l e c t r i c a l shadow' c a s t on the ground by the t a l l  tower s t r u c t u r e w i t h power l i n e s . the  lines  The frequency  of l i g h t n i n g  i s assumed t o be the frequency  s t r o k e s on  o f s t r o k e s to the  The w i d t h (w) of the shadow a r e a e s t i m a t e d by r e f e r e n c e 6 i s  F o r a power l i n e w i t h two ground w i r e s , the w i d t h i s g i v e n by  (see F i g u r e  1.7)  w = 4h + b  where  h = h e i g h t of ground w i r e i n m b = s e p a r a t i o n between ground w i r e s  Similarly,  f o r a power l i n e w i t h o n l y 1 ground w i r e , the w i d t h i s g i v e n  by  w = 4h  where  and  h = h e i g h t of ground w i r e i n m  f o r power l i n e s without  ground w i r e s , the width i s g i v e n by  w = 4h + b  where  h = h e i g h t of phase w i r e  in m  b = s e p a r a t i o n between outermost phase w i r e s  Thus, the number of strokes/km - y r to the power l i n e  N  L  = 0.1  (IKL)  s t r o k e /km - y r  (N ) i s Li  (1.1)  13  F o r a t y p i c a l 500 kV tower o f the MICA Dam c l o s e t o the s u b s t a t i o n , we  fr\  AT  L  =  Project,  f o r the l i n e  have  i w o r > \  (O- ^ 1  3 0  4  x  37.5  )  +  18.64  loOO  = 0.5 s t r o k e s /km  GW  ( 1  '  2 )  - yr  GW  O PW  GW PW  :  = ground w i r e = phase w i r e  M .  2h  b  2h $  s  = s h i e l d i n g angle = 23°  h  = h e i g h t of ground = 37.5  wire  m  b = width between ground = 18.64  F i g u r e 1.7:  6.  wire  m  L i g h t n i n g s t r o k e ' E l e c t r i c a l shadows' of a t y p i c a l 500 kV t r a n s m i s s i o n l i n e .  S h i e l d i n g f a i l u r e phenomenon of l i g h t n i n g As shown i n F i g u r e 1.7,  strokes  ground w i r e s a r e designed f o r s h i e l d i n g o f  the phase w i r e from d i r e c t l i g h t n i n g s t r o k e s .  However, l i g h t n i n g  strokes  14  could  still  'sneak' through the ground wire and  s h i e l d i n g f a i l u r e s have been r e c o r d e d  h i t the phase w i r e .  Such  in various countries for different  tower conf i g u r a t i o n s . Maikopar^ d e r i v e d a s h i e l d i n g f a i l u r e curve data  (see F i g u r e 1.8).  However, t h e graph does not  s h i e l d i n g f a i l u r e s occur m a i n l y on lower l i g h t n i n g  based on observed shows t h e f a c t  At  is effectively  strokes.  As seen from F i g u r e 1.1 from the t h u n d e r c l o u d  that  strike currents.  h i g h e r c u r r e n t s , (e.g. >14.2;kA f o r MICA), t h e phase w i r e s h i e l d e d from l i g h t n i n g  field  and  1.2,  i s formed and  t h e p i l o t downward stepped  propagates earthward f r e e l y  of the s t r u c t u r e on e a r t h i n i t i a l l y .  leader  regardless  L a t e r , t h e r e t u r n s t r o k e s i s formed  from a ground o b j e c t c l o s e s t to the l e a d e r t i p , (ground w i r e , phase w i r e , or the ground) arid propagates upward t o meet t h e stepped l e a d e r to the l i g h t n i n g path. by the l i g h t n i n g  T h i s ground o b j e c t  i s the o b j e c t which w i l l be  complete struck  stroke.  8 Brown  analysed  f i n d e r P r o j e c t and  r e s u l t s from t h e 120,000 km  the s t r i k i n g d i s t a n c e r current  i n the  deduced t h a t t h e t a r g e t i s not chosen u n t i l the  between the stepped l e a d e r t i p and  the s t r o k e  - yr l i n e  .  the prospective object  r  g  7 i T 0•75 = 7.1 I  distance  i s s h o r t e r than  This s t r i k i n g distance i s related  as  Path-  only to  m  where I = current  i n kA  From t h i s s t r i k i n g d i s t a n c e concept, we geometric model, as  shown i n F i g u r e 1.9a.  The  can d e v e l o p  electro-  s h i e l d i n g f a i l u r e of ground-  w i r e a t lower c u r r e n t a m p l i t u d e s can c o r r e c t l y e x p l a i n e d r e f i n e d method. The degree  the  by t h i s more  of exposure of d i f f e r e n t c o n d u c t o r s i s  15  SHIELD ANGLE - DEGREES  F i g u r e 1.8:  P r o b a b i l i t y of S h i e l d i n g F a i l u r e v s . S h i e l d A n g l e between Ground Wire and Top Phase C o n d u c t o r /  r e p r e s e n t e d by drawing exposure a r c s o f s t r i k i n g d i s t a n c e r a d i u s , and c e n t r e d a t each i n d i v i d u a l c o n d u c t o r s . The i n i t i a l power f r e q u e n c y v o l t a g e of t h e phase conductor i s ignored a s t h i s v o l t a g e i s c o m p a r a t i v e l y s m a l l to t h e d i s c h a r g e v o l t a g e of t h e l i g h t n i n g For  lightning  c u r r e n t s of 10 kA and  exposure of t h e phase conductor PW conductor exposure t o l i g h t n i n g  strokes.  14.2  kA,  the  i s shown. I t can be seen t h a t t h e phase  s t r o k e i s decreased w i t h i n c r e a s e s i n  s t r i k i n g c u r r e n t . For c u r r e n t of amplitudes h i g h e r than 14.2 tower  corresponding  kA, f o r t h i s  s t r u c t u r e i n MICA p r o j e c t , the phase c o n d u c t o r i s e f f e c t i v e l y  s h i e l d e d by t h e ground w i r e and t h e ground a s t h e exposure a r c i s negligible in size  (See F i g u r e 1.9a).  16  BC=phase w i r e exposed a r c for shielding f a i l u r e SI  F i g u r e 1.9a: E l e c t r o g e o m e t r i c model w i t h maximum s t r i k i n g d i s t a n c e of 53.3m.  Stroke Current in kA  F i g u r e 1.9b:  Frequency D i s t r i b u t i o n o f S h i e l d i n g F a i l u r e S t r o k e C u r r e n t s i n case of S h i e l d i n g F a i l u r e  17  Brown et a l f u r t h e r  investigated  a n g u l a r d i s t r i b u t i o n of t h e l i g h t n i n g  t h i s s i t u a t i o n by t a k i n g t h e  s t r o k e g(^) i n t o account and  e v a l u a t e d t h e phase w i r e exposed a r c f o r d i f f e r e n t s t r o k e c u r r e n t s as  x  =  r  s i n i i ^ l  h  ^  2 2 g(^) = - c o s y  where:  The  d e t a i l e d a n a l y t i c a l r e s u l t s a r e shown i n F i g u r e 1.9b.  MICA tower o f maximum s t r i k i n g d i s t a n c e of shows t h a t  (1.4)  l e s s than 1% o f s h i e l d i n g  w i r e w i l l exceed 14 kA.  For  our  53.3 m ( I =14.2kA), t h e r e s u l t  f a i l u r e l i g h t n i n g c u r r e n t s t o t h e phase  T h i s a g r e e s w e l l w i t h t h e geometric  interpretation  shown i n F i g u r e 1.9a. The  lightning  stroke u s u a l l y  t h i s case, a v o l t a g e w i l l potential  h i t t h e ground w i r e o r t h e tower. In  b u i l d up ."across t h e i n s u l a t o r because of t h e  r i s e on t h e tower crossarms. I f t h e i n s u l a t o r  f l a s h over  (' b a c k f l a s h o v e r ' ) t o t h e phase c o n d u c t o r s , then l i g h t n i n g surges w i l l appear on t h e c o n d u c t o r s .  overvoltage  18  CHAPTER 2 :  1.  LIGHTNING SURGE PROPAGATION IN OVERHEAD TRANSMISSION LINES  Introduct ion Propagation  overs ments.  of l i g h t n i n g  surges  due  to d i r e c t  strokes, or b a c k f l a s h -  i n overhead l i n e s i n f l u e n c e s the c h o i c e o f i n s u l a t i o n One must know the a t t e n u a t i o n and  require-  d i s t o r t i o n c h a r a c t e r i s t i c s of  the  l i n e i n o r d e r t o f i n d t h e o v e r v o l t a g e s e n t e r i n g t h e s u b s t a t i o n where most of the equipment i s c o n c e n t r a t e d .  T h i s s e c t i o n t r i e s to answer the  whether i t i s p o s s i b l e to r e p r e s e n t untransposed equivalent  overhead l i n e s  s i n g l e phase l i n e s f o r the s t r i c k e n conductor  and whether s e l f , p o s i t i v e or zero such s i n g l e - p h a s e At f i r s t ,  with  questions  as  accuracy,  sequence impedances should be used i n  representations? f i e l d t e s t s r e s u l t s a r e d u p l i c a t e d by u s i n g a F o u r i e r 9  t r a n s f o r m a t i o n method.  T h i s method not o n l y i n c l u d e s the  dependence of the l i n e parameters, but  i t a l s o u s e s t h e exact  frequency-dependent t r a n s f o r m a t i o n m a t r i x which r e q u i r e s at each frequency (e.g. 10 k Hz  to  w i t h i n the frequency 1MHz).  frequencycomplex,  recomputation  r a n g e . t y p i c a l of l i g h t n i n g  surges  T h i s method i s recommended f o r t h e s i m u l a t i o n  of d i s t a n t s t r o k e s where the frequency  dependent c h a r a c t e r i s t i c s must  be  included. For c l o s e - b y l i g h t n i n g  s t r o k e s , t h e above frequency-domain  can be r e p l a c e d by a s i m p l e r time-domain s o l u t i o n method. i s based on modal a n a l y s i s w i t h frequency-independent valued transformation matrices.  The  T h i s method  parameters and  r e s u l t s o b t a i n e d w i t h the  time-domain s i m u l a t i o n method agree v e r y w e l l ( < 4% d e v i a t i o n ) the a c c u r a t e  frequency-domain s i m u l a t i o n method.  solution  real-  simpler with  A f t e r c o n f i r m i n g t h e c o r r e c t n e s s i n the time-domain  simulation  w i t h the exact N - phase r e p r e s e n t a t i o n of t h e overhead l i n e f o r c l o s e - b y l i g h t n i n g s t r o k e s , the r e s u l t s o b t a i n e d  are thus compared a g a i n s t  phase approximate r e p r e s e n t a t i o n s as p r e s e n t l y used. a d d i t i o n a l recommendations a r e made on how  Furthermore,  to remove u n c e r t a i n t i e s i n the  c h o i c e of surge impedance v a l u e s of overhead l i n e s . ^ t h a t frequency  dependence e f f e c t of nearby l i g h t n i n g  L i n e parameters can be chosen a t h i g h f r e q u e n c y  Modal a n a l y s i s f o r N - phase u n t r a n s p o s e d The  coupled  easily  and  ignored, line  line d e s c r i b e the  waves on overhead t r a n s m i s s i o n l i n e s .  the s i n g l e phase case, the s o l u t i o n to t h e N-  obtained  s t r o k e can be  e.g. a t 1 M Hz,  w e l l known t r a n s m i s s i o n l i n e e q u a t i o n s  of e l e c t r o m a g n e t i c to  I t i s a l s o found  11 12 ignored as c o n t r a d i c t o r y to the p r e v i o u s f i n d i n g s . '  r e s i s t a n c e can be  2.  single-  propagation  However, c o n t r a r y  phase case cannot  be  s i n c e each of t h e N overhead c o n d u c t o r s i s m u t u a l l y  to the o t h e r c o n d u c t o r s .  The  f o l l o w i n g two  second-order p a r t i a l d i f f e r e n t i a l m a t r i x i n v o l t a g e s and  currents along dV dx  J  dl dx  equations  s e t s of  d e s c r i b i n g t h e change  the N - phase l i n e must be  phase"  phase  r  nxl  "  phase" =  1  J  r P  h a s e  Y  L  r  nxn  i J  nxl  J  rv L  solved:  phase  L  nxn  simultaneous  nxl  p h a s e J  i nxl  where [z  p h a s e  ].  nxn [ phase-| nxn [-,-phase j nxl phase. [V nxl Y  J  impedance m a t r i x admittance m a t r i x  i n phase domain i n phase domain  phase c u r r e n t  vector  phase v o l t a g e  vector  (2.1)  (2.2)  20 The N coupled d i f f e r e n t i a l equations i n equations (2.1) and (2.2) can.be transformed into N decoupled equations by replacing phase quant i t i e s with modal quantities,  [ v  phas  [I  phase  e ]  ]  =  [ T  ]  =  [ T  _  [v  3  mod  [I  (2.3)  e]  mode  (2.4)  ]  and by choosing [T ] and [ T ] i n a certain way, as described l a t e r . ±  Applying  equations (2.3) and (2.4) to equations (2.1) and (2.2) gives demode dx  [T,]"  [zP  1  h a S e  ]  [T.]  [I ° m  d e  (2.5)  ]  _ j-^mode^ |.^.mode-|  (2.6)  and dl dx  mode  = [I.]"  =  [ P  1  h a S e  Y  ]  [T ] [ . V  node  ]  j-mode^ ^ o d e ^  (2.7)  (2.8)  Y  To find ^T ], we f i r s t d i f f e r e n t i a t e equation (2.1) with respect to *" phase] dl with equation (2.2): x , and replace dx V  1  dV dx  h  a  s  e  l  ^phase-j j. phase^ Y  ^phase^  (2.9)  2  With equation (2.3), t h i s can be written i n modal quantities as 2„mode  = ty"  1  [Z  p h a s e  ]  [Y  p h a S e  ]  [T ] 1^°**] v  (2.10)  dx (2.11)  If  [T 1 i s t h e m a t r i x of e i g e n v e c t o r s o f I Z v  ]  p h a S e  [y  p h a s e  ],  then [ A ]  becomes a d i a g o n a l m a t r i x , w i t h i t s elements b e i n g t h e e i g e n v a l u e s o f j- phase^  ^phase^  z  Similarly,  f o r t h e c u r r e n t q u a n t i t i e s , we have  2 mode a L_ J  T  dx  L  =  [T.]  -  UJ  [Y  - 1  P  H  A  S  ]  £  [Z  p h a S e  ]  [T.] [ I ° m  d e  ]  (2.12)  2  [I ° m  d e  ]  (2.13)  where [ T \ ] = m a t r i x o f e i g e n v e c t o r s o f [Y*\ being i d e n t i c a l  to t h a t i n e q u a t i o n  ] [7?  ], with [ A ]  (2.11).  T a k i n g t h e t r a n s p o s e of t h e e x p r e s s i o n f o r [ A ] i n e q u a t i o n and that  comparing i t w i t h t h a t f o r [ A ] i n e q u a t i o n [Z ^ p  a s e  ]  and [ y P ^  a s e  ]  [Z  ±  or  IT ]  _  p h a S e  tZ  1  v  [ T ] = (-[T.] ")" 1  ]  P h a S £  [Y  ]  P  [Y  , [rT ^ J - m a t r i x , we •  m -i  from e q u a t i o n  m  d e  P  S  H  ]  £  A  S  ([T.]^"  1  ] [T ]  E  V  (2.14)  1  i.i  J  M  D E  ] = [7 f  [Z  ]  p h a S e  ]  i  <r  [T.]  Using only  rr,niode,  . ,,mode, J and [ Y J r  (2.15)  (2. 8) as  = [I.]  [ P  - 1  Y  h a S e  ] ([T.] )-  mode,-! _ t phase,-l [Y ] = [T.] [ Y ] r  o r [T ] i s needed.  can o b t a i n t h e modal parameters o f [Z  ±  from e q u a t i o n  [Y °  or  A  (2.6) as  [Z °  and  H  v  Thus, o n l y one of t h e m a t r i c e s  the  remembering  a r e symmetric, g i v e s :  [A] = [? f  =  (2.10), w h i l e  (2.12)  r v  1 1  r  , [T.]  m  1  (2.15a)  (2.16)  In the computer pugram developed equation  (2.16) i s used  i n v e r s e of [T ] and from which [ Y k p  a s e  f o r these two  f o r t h i s modal a n a l y s i s , reasons:  i t does not  s e c o n d l y , the program c a l c u l a t e s  ] i s o b t a i n e d by  inversion.  [Y  m o d e  [Y ]  13  (2.15), but  (2.16).  i n a s i m p l e r way  [Z ° m  d e  ] =  [A] [ Y  [Z ° m  d e  14  r e q u i r e the ]  i s then  first  ] i s not c a l c u l a t e d from  anyhow,  easily  o b t a i n e d by t a k i n g the r e c i p r o c a l of the d i a g o n a l elements of the hand s i d e of e q u a t i o n  '  right-  equation  from  m o d e  ] ^  (2.17)  -  t h a t i s , each component i s simply „mode i  Y  l mode  T h i s i s v a l i d because  [A] from e q u a t i o n  IA] =  [T^"  1  =  [T,]"  1  = [Z  3.  (  m o d e  [ P  h a s e  Z  [Z  J.  p h a S e  [Y ° m  h a S e  [T.].  d e  are o n l y determined  ]  [ T J  l  g  )  [T.]"  1  .  [YP  h a S e  ]  [ T J  ]  (2.19)  R o t a t i o n o f e i g e n v e c t o r s f o r z e r o shunt I t has to be noted  <  (2.11) can be r e w r i t t e n as  ] [YP  ]  2  conductance  t h a t the e i g e n v e c t o r s  (columns of [T^] or  to w i t h i n a m u l t i p l i c a t i v e constant.  [T ]) y  Each e i g e n v e c t o r  can, t h e r e f o r e , be m u l t i p l i e d w i t h any non-zero complex s c a l a r , and  i t will  s t i l l be the c o r r e c t e i g e n v e c t o r . S i n c e we  assume zero phase shunt  d i s c u s s e d l a t e r i n Chapter  conductances (corona l o s s e s w i l l  4 ) , the modal conductances should a l s o be  be  zero.  T h i s can be a c h i e v e d by m u l t i p l y i n g t h e e i g e n v e c t o r s w i t h a p r o p e r l y chosen constant.  Then e q u a t i o n  ( 2 . 8 ) , which i s d e f i n e d i n the frequency  domain,  23  can be r e w r i t t e n i n the time domain as f o l l o w s  8.mode 1  =  3x  [ c  m o d e  3 modd v 9t  ]  In order t p o b t a i n zero modal conductances, a r o t a t i o n scheme i s used which makes t h e modal admittance m a t r i x  L  mode  J  m =  rotate  { L  } J  j [ B J L  °  d  [Y  m o c  ^ ] purely e  imaginary,  ] ^ rotate  e  J  This r o t a t i o n i s equivalent to d i v i d i n g the i - t h eigenvector of  [T ]) by a f a c t o r D^.  First,  f i n d t h e angle  6^ o f Y ^  m o d e  ( i - t h column  , as shown  i n F i g u r e 2.1. Then 90° - 0. (2.20)  D. = e I  With a l l Dj^'s forming  a diagonal matrix  [ d ] , the m o d i f i e d m a t r i x o f  e i g e n v e c t o r s becomes  IT.] = [T.] i rotate • i 1  J  JL  Then from e q u a t i o n  [Y °  d  [Y °  d  m  e  J  (2.21)  [D]  (2.15a),  ]  r  o  t  a  t  e  = [D] [ T . ] "  ]  r  o  t  a  t  e  = [D] [ Y °  1  [YP  h a S e  ]  ([T.] )" 1  1  [D]  (2.22)  or  m  e  m  Since a l l matrices  d e  i n equation  simply a t o t a t i o n of Y ° m  d e  (2.23) a r e d i a g o n a l , e q u a t i o n  by an angle  „ , , „mode, F i g u r e 2.1, makes [Y ^rotate P r  (2.23)  ] [D]  ,  u r e l  y  (2.23) i s  (90° - 6 ^ , which a c c o r d i n g t o imaginary.  24  [Y ° 0 m  After  1  from e q u a t i o n n  d e  i s found  from e q u a t i o n  rotate  (2.21),  [Z °  mode  m  H  d e  ]  rotate  (2.23), and [T.] ' i rotate 1  i s calculated  from  mode rotate 1  [Y  rotate  These modal q u a n t i t i e s and t r a n s f o r m a t i o n m a t r i c e s o b t a i n e d a r e c h a r a c t e r i s t i c s of the p a r t i c u l a r d e s i g n of t h e untransposed  line.  These modal parameters and modal t r a n s f o r m a t i o n m a t r i c e s a r e needed as i n p u t f o r t h e r e p r e s e n t a t i o n o f untransposed  d i s t r i b u t e d - parameter  lines  i n t h e time domain s o l u t i o n , such as i n t h e UBC v e r s i o n o f t h e E l e c t r o magnetic T r a n s i e n t s Program as d e s c r i b e d i n  F i g u r e 2.1:  Complex Y  m o d e  13 14 '  b e f o r e and a f t e r  rotation  25  4.  C o n f i r m a t i o n o f accuracy o f e i g e n v a l u e and e i g e n v e c t o r s u b r o u t i n e The UBC Computing Centre l i b r a r y s u b r o u t i n e DCEIGN^^  compute the e i g e n v a l u e s and engenvectors double p r e c i s i o n s u b r o u t i n e f i r s t m a t r i x H.  i s chosen t o  o f the [ Y ] - [ Z ] m a t r i x .  reduces  This  the complex m a t r i x t o a Hessenburg  The s u b d i a g o n a l elements of H a r e then f o r c e d t o converge t o 49  zero by the m o d i f i e d LR method. to the e i g e n v a l u e s .  Hence the d i a g o n a l elements o f H converge  The e i g e n v e c t o r s can then be o b t a i n e d by backward  substitution. The  c o r r e c t n e s s of t h e program has been checked  by comparing i t s  output w i t h p u b l i s h e d r e s u l t s f o r a d o u b l e - c i r c u i t line"*"^.  Both r e s u l t s  of modal a t t e n u a t i o n s and modal v e l o c i t i e s agree t o w i t h i n t h r e e (see Table 2 . 1 ) .  The modal m a t r i c e s  [T ] d i f f e r o n l y s l i g h t l y  digits  (see T a b l e  2.2) . T a b l e 2.1 UBC & BPA modal a n a l y s i s r e s u l t s f o r a 735 kV l i n e Modal a t t e n u a t i o n  neper/mile BPA  5.  16  Modal v e l o c i t y UBC  miles /s c  BPA  .15998E6  .15998E6  .61227E-1  .612E-1  .18438E6  .18437E6  .19050E-2  .191E-2  .18497E6  .18497E6  .18209E-2  .182 E-2  .18606E6  .18605E6  .54529E-3  .544E-3  .18615E6  .18614E6  .50169E-3  .502E-2  .18614E6  .18614E6  .47704E-3  .475E-3  R e a l - v a l u e d frequency  - independent  transformation matrix  Time domain s o l u t i o n s w i t h t h e t r a n s f o r m a t i o n m a t r i x difficult  [T_^] become  i n t h e o r y s i n c e [T^] i s complex as w e l l as f r e q u e n c y - dependent.  Table  UBC and BPA modal m a t r i x  . 3412-j.0022  .5558+j.0  .3948-j.0157  •3324+j.0230  ,4822-j.0  -.3128+J.0248  . 3412-j.0022  .5558+j.0  .3948-j.0157  .3324+j.0230  2.2  [T ] r e s u l t s f o r a 735 kV l i n e  -.4959-j.0262 .5486-j.0 -.1118+J.0056 -.4959-j.0262 .5486-j.0  .1730-j.0017  .3209+j.0008  .4647-j.0247  .4804+j.O  .5410-j.0  -.4145+j.0304  -.1730+J.0017  -.3209-j.0008  -. 4647+j.0247  -,4804+j.0  .6827+j.0 -.3550-j.0021 •0812+j.0061 -.6827+j.0 .3550+j.0021  .4145-J.0304  -.0812-j.0061  .1730-j.0298  .3209+j.0208  .6827+j.0737  .5453-j.0784  .4681-J.0598  .4659+j.0285  -.3637-j.0422  -.3153-j.0095  -.1105+j.0215  .5469-j.0408  -.4064+J.0048  .0869+j.0159  .3412-j.0022  •5558+j.0612  -.4959+j.0452  -.1730+J.0298  -.3209-j.0208  -.6827+j.0737  .3955-j.0157  .3294+j.0594  .5453-j.0784  -.4681+j.0598  -.4659-j.0285  •3637-j.0422  -.3153-j.0095  -.1105+j.0215  -.5469+j.0408  .4064-j.0048  -.0869+j.0159  .4822-j.0  -.3128+J.0248  -.1118+j.0056  . 3412-j.0022  •5558+j.0612  -.4959+j.0452  .3955-J.0157  .3294+J.0594  .4832-j.0  .4832-j.0  -,5410+j.0  27  However, the imaginary p a r t o f the m a t r i x [T ] i s always s m a l l compared w i t h i t s r e a l p a r t . the m a t r i x i t s e l f , we enough ( ^ 2%  5%)  By t a k i n g the r e a l p a r t o r the magnitude of  o b t a i n modal parameters  which-are • s t i l l  accurate  deviation).  Furthermore,  the a t t e n u a t i o n caused by corona may  be much h i g h e r  than t h a t caused by the s e r i e s r e s i s t a n c e and f o r c l o s e - b y s t r o k e s , t r a n s m i s s i o n l i n e s s h o u l d be r e p r e s e n t e d as l o s s l e s s .  With the  approximations,  the frequency dependence of the modal t r a n s f o r m a t i o n m a t r i x d i s a p p e a r s . It  i s t h e r e f o r e recommended t h a t the complex m a t r i x be approximated  r e a l - v a l u e d , frequency-independent e a s i e r f o r two a)  matrix.  by a  T h i s makes s i m u l a t i o n s much  reasons:  A frequency independent  modal m a t r i x does n o t r e q u i r e recompu-  t a t i o n o f the modal m a t r i x a t each frequency c o n s i d e r e d w i t h i n the l i g h t n i n g frequency range, e.g. b)  A r e a l - v a l u e d modal m a t r i x enables d i r e c t to  6.  10 kHz  Frequency  be performed  t o 100  kHz.  transient simulation  i n t h e time domain.  dependent e f f e c t s i n l i g h t n i n g surge p r o p a g a t i o n  To i n c l u d e frequency dependent e f f e c t s i n t r a n s i e n t o v e r v o l t a g e 17 s t u d i e s i s a c o m p l i c a t e d t o p i c by i t s e l f .  Meyer, Dommel  18 and M a r t i  have i n v e s t i g a t e d the t ime domain methods u s i n g c o n v o l u t i o n i n t e g r a l s weighting functions.  However, the frequency domain s o l u t i o n s can a l s o  and be  Q  o b t a i n e d by the F o u r i e r T r a n s f o r m a t i o n methods. domain method i s inadequate  to account  Though the  frequency  f o r the n o n - l i n e a r phenomenon  (e.g.  corona d i s c h a r g e ) and the time domain phenomena (e.g. i n s u l a t o r  back  —  f l a s h o v e r or a r r e s t e r o p e r a t i o n ) , i t i s s u f f i c i e n t  f o r the  purpose  28 of  s t u d y i n g frequency dependent e f f e c t s on l i g h t n i n g surge p r o p a g a t i o n i n o v e r -  head  lines. 9 As d i s c u s s e d i n an e a r l i e r work  ,  the  frequency  i n c l u d e s frequency dependence of l i n e parameters.  domain s o l u t i o n s  I t a l s o uses the exact  complex frequency dependent t r a n s f o r m a t i o n m a t r i x to be computed at  each  frequency p o i n t , and employs the l i n e a r i n t e r p o l a t i o n t e c h n i q u e i n e v a l u a t i n g the F o u r i e r T r a n s f o r m a t i o n i n t e g r a l s . 19 by Ametani km  The r e s u l t s from a measured f i e l d  test  20 '  of a l a b o r a t o r y generated d i s t a n t l i g h t n i n g wavefront  from the s u b s t a t i o n was  83.212  s u c c e s s f u l l y d u p l i c a t e d by the author u s i n g t h e  F o u r i e r T r a n s f o r m a t i o n method. (See F i g u r e 2.2).  Due  pendent e f f e c t of the l i n e parameters,  r i s e time of 2 ys of the  wavefront now line.  increased  an i n i t i a l  to the frequency  de-  t o about 40 ys as the wave t r a v e l l e d down the  Thus, the frequency dependent e f f e c t must be i n c l u d e d f o r th§  d i s t a n t l i g h t n i n g s t r o k e case.  The l i g h t n i n g waveshape o b t a i n e d a f t e r  the s t r o k e has t r a v e l l e d from the s t r i k i n g p o i n t to the s u b s t a t i o n then be i n t e r f a c e d w i t h the time domain s o l u t i o n s as used 47 magnetic t r a n s i e n t s program. For  close-by l i g h t n i n g strokes,  the  i n an  electro-  r e s u l t i n g waveshapes  can a g a i n be o b t a i n e d by the F o u r i e r T r a n s f o r m a t i o n i n t e g r a l s , and s i m p l e r time domain methods. untransposed l i n e can be f i r s t independent  parameters  in previous s e c t i o n s ) .  For the time domain method,- the m u l t i - p h a s e s o l v e d by modal a n a l y s i s u s i n g frequency  Then, t h i s m u l t i - p h a s e l i n e i s r e p r e s e n t e d by a s i n g l e As shown i n F i g u r e 2.3,  these methods agree q u i t e w e l l  (< 4% d e v i a t i o n ) .  r e p r e s e n t a t i o n w i t h frequency independent  lightning  the  and r e a l - v a l u e d t r a n s f o r m a t i o n m a t r i x (as d e s c r i b e d  phase l i n e a p p r o x i m a t i o n .  by s t r o k e case because  can  variations  effect  r e s u l t s o b t a i n e d by a l l The  s i n g l e phase l i n e  is valid  in this  close-  among the modal a r r i v a l times a t range of  f r e q u e n c i e s are not apparent  i n such s h o r t d i s t a n c e s (e.g. < 2  km).  Voltages(p.u) 3<|> w i t h frequency dependence 3<j> without frequency dependence without frequency dependence  ime  F i g u r e 2.3:  Close-by l i g h t n i n g s t r o k e case s o l v e d by frequency and time domain methods.  30 t=0  83.212 km A B • C  Output  voltage (p.u.) f i e l d measurements 3<j> w i t h frequency dependence  1.0  0.8  0.6  •0.4  0.2  -0.2  F i g u r e 2.2:  Numerical s i m u l a t i o n o f o v e r voltage taking untransposition and frequency dependence i n t o account.(Ref. 9)  31  In such a r e p r e s e n t a t i o n , s e l f  impedance  higher frequencies  should be used to approximate t h e  (e.g. 1 MHz)  dependance c h a r a c t e r i s t i c s of the However,  of one  taken i n choosing  The  phase f o r a t y p i c a l 500  frequency  higher  The  kV l i n e i s shown i n Table 2.3.  r e s i s t a n c e to r e a c t a n c e  f r e q u e n c i e s , e.g.  2.8%  line resistance for  dependence of l i n e  that the a t t e n u a t i o n of the wave i s n e g l i g i b l e f o r 1 km.  frequency  line.  c a u t i o n must be  the l i g h t n i n g surge s t u d i e s .  - parameters c a l c u l a t e d at  (< 5%)  parameters I t i s shown  up to about 100  kHz  r a t i o i s also small e s p e c i a l l y at  at 1 M Hz.  Furthermore, s i n c e the  Bergeron's  method of c h a r a c t e r i s t i c i n s o l v i n g the t r a n s m i s s i o n l i n e e q u a t i o n  is valid  o n l y f o r a l o s s l e s s t r a n s m i s s i o n l i n e , d i s t r i b u t e d l i n e l o s s e s are u s u a l l y approximated by  lumping the r e s i s t a n c e at c e r t a i n l o c a t i o n s .  r e s i s t a n c e at 1 M Hz may  cause i n a c c u r a c y  i n the s i m u l a t i o n .  This On  high  the  other  hand, surge impedances c a l c u l a t e d by  z  surge  A  =  + J^L  (  where R/jwL = 2.8% z  surge  a t 1 M Hz  (See T a b l e  fcjL  =  2  >  2  5  )  2.3)  ( 2 > 2 6 )  / JUJC  are e s s e n t i a l l y i d e n t i c a l f o r t h i s l o s s y and p l i c a t e d frequency ignored, and acceptable  l o s s l e s s cases.  Thus, com-  dependent e f f e c t s f o r the nearby s t r o k e case can  . frequency  accuracy(.,,See  independent and F i g u r e 2.9  be  l o s s l e s s representation give  ).  ./  . Therefore,.the  p r e v i o u s methods of m o d e l l i n g  • -i j • e x p o n e n t i a l decay i n o v e r v o l t a g e s  a r e not a c c e p t a b l e . techniques,  1  2  They should be  or any  •  l i n e l o s s e s by i  r e s i s t a n c e lumping scheme  r e p l a c e d by d e t a i l e d w e i g h t i n g  or F o u r i e r T r a n s f o r m a t i o n  simple 11,21  function  methods f o r d i s t a n t s t r o k e , or by  32  Frequency (Hz)  Resistance  Reactance  R(fi/km)  X(ft/km)  R/X %  z  surge  velocity  ( a )  (m/ys)  Attenuation e" ^( Y  in  6  183.  6525.  2.8  291.  280.  .73  io  5  42.  692.  6.  300.  272.  .93  io  4  7.  78.  9.  317.  257.  .989  io  3  .9  10.  340.  240.  .999  Table  8.9  2.3: Frequency dependence of s e l f q u a n t i t i e s of parameters f o r a 3 phase 500 kV l i n e .  line  /km)  33  lossless  7.  l i n e r e p r e s e n t a t i o n f o r a nearby s t r o k e as d e s c r i b e d  i n above,  D e t e r m i n a t i o n o f t h e surge impedance o f the s t r u c k phase of a transmission l i n e . An a c c u r a t e and r e l i a b l e v a l u e of the surge impedance i n phase domain  must be o b t a i n e d a)  as due t o the f o l l o w i n g  The amount o f o v e r v o l t a g e  reasons:  wave t r a n s m i t t e d from t h e overhead  l i n e t o t h e underground SF^ c a b l e a t t h e c a b l e - l i n e determined by the surge impedances o f d i f f e r e n t refraction  junction i s  components.  coefficienttC_ i s R cable" C , surge l:i.ne ^cable surge surge 2  The  Z  R  z  7  )  +  where ^cable _ surge  g  u  line _ surge  g  u  z  b)  The exact v a l u e  i p d  e  m  e  a n  ce  of cable  impedance o f l i n e (e.g. 304 ft)  r  o f the o v e r v o l t a g e  from the l i g h t n i n g  (e.g. 60 ft)  stroke  wave on the l i n e , r e s u l t i n g  (1^) i s d i r e c t l y r e l a t e d  t o surge  -i . r,line impedance o f the l x n e Z as surge r  v  =  ^  2  .  l z  i  n  (2.28)  e  surge  T h i s r e s u l t i n g o v e r v o l t a g e wave impresses e l e c t r i c a l s t r e s s on e x t e r n a l and i n t e r n a l i n s u l a t i o n concern i n t h e i n s u l a t i o n  o f the system and forms the main  co-ordination  study.  34  In  s p i t e of the above important c r i t e r i a , u n c e r t a i n t i e s i n surge  im pedance c a l c u l a t i o n s o f overhead give r e l a t i v e l y  l i n e do e x i s t . ^ ' ^ ' ^  Reference  11  lower surge impedance r e s u l t s f o r the ground w i r e (352 ft) 12  as compared t o D a r v e n i z a ' s computation. for  D a r v e n i z a c l a i m s t h a t the e q u a t i o n  surge impedance i n phase domain as i s g i v e n by: Z  *f self S  U  Z \ mutual S U r g €  where  = 60 In ^ r  "  = 60 In ^b..  (2.29)  (2.30)  h = conductor h e i g h t ;  r  = conductor r a d i u s  a. .. = s e p a r a t i o n between conductors ij b. . = s e p a r a t i o n between conductor and ij o t h e r conductor image  T h i s i s r e a d i l y d e r i v e d from the p o t e n t i a l c o e f f i c i e n t P and the i n d u c t a n c e term L a s :  5  seif • h r r self J  L  and  1  self  Z ^ f self  = Ho2TT  =  to  ^r-  "  l  n  T  ^h r  / ^ - - / C ... self  < - > 2  (  A  __P = 60 £ n ^ self self ~  2  -  3I  3  2  )  (2.33)  35  where  u  = permeability -7 = 4TT x 10 H/m  °  e = permit i v i t y ° 1 -9 = x 10 F/m 36TT  [C] = [ P ]  _ 1  [P] = p o t e n t i a l c o e f f i c i e n t m a t r i x , w i t h d i a g o n a l term  P self  [L] = i n d u c t a n c e m a t r i x , w i t h d i a g o n a l term °  L self  However, the above formulae n e g l e c t • Carson's c o r r e c t i o n terms, o t h e r c o n d u c t o r s , and ground w i r e s calculation  used  for  earth return.  A detailed  f o r t h e surge impedance m a t r i x i n phase domain [Z ] surge &  r  v  must be performed t o i n o r d e r t o j u s t i f y t h i s I f we  assumption.  c o n s i d e r the r e l a t i o n s h i p between the surge impedance  m a t r i x i n both phase and modal domain as  [ y  and  phase  ]  [V °  ]  m  d e  then by s u b s t i t u t i n g  [T  or  [V  [ z  phase  36  p h a S e  ]  ] [ ; I  phas  = [Z ° ][ I ° surge m  eqs.(2.3)  TV ** ] 1  v  =  d e  m  d e  ( 2  ]  d e  _  1  [ l  (2.35), we  p  h  a  S  e  = [T ] [ Z ° ][T.]- [l v surge I m  d e  _  3 4 )  (2.35)  & (2.4) i n t o  = [Z ° ][T ] surge I m  e ]  1  ]  p h a S e  can get  (2.36)  ]  (2.37)  36  Comparing e q u a t i o n s (34) and  [Z  (37) , we  thus o b t a i n  -1 phase .mode J[T.] ] = [T ] [ Z 'surge surge  (2.38)  The above r e l a t i o n i n e q u a t i o n (2.38) i s i d e n t i c a l t o t h a t d e r i v e d by Wedepohl.  22  phase c u r r e n t i s f i r s t  In h i s method, the r e f l e c t i o n c o e f f i c i e n t f o r obtained.  o b t a i n the e x p r e s s i o n f o r [ Z ^ P  a s e  The ]  coefficient  i s then s e t t o zero to  as i n e q u a t i o n (2.38).  surge  M  R e s u l t s f o r the surge impedance from e q u a t i o n s f o r both the ground  (2.29) and  and the phase w i r e s are shown i n T a b l e 2.4.  seen from the t a b l e ,  (2.38)  As can be  the surge impedance o b t a i n e d by D a r v e n i z a ' s  formula  which n e g l e c t s the s k i n e f f e c t o f the e a r t h r e t u r n component i n t r o d u c e s negligible deviation  (about 1%).  However, the D a r v e n i z a ' s f o r m u l a s h o u l d  o n l y be used when the ground w i r e i s t r e a t e d as another i n d i v i d u a l phase (e.g. f o r the c l o s e - b y l i g h t n i n g s t r o k e c a s e ) . w i r e as another component f o r e a r t h r e t u r n  I f one takes the  ground  (e.g. f o r the d i s t a n t s t r o k e  c a s e ) , the formula f o r s e l f surge impedance must be m o d i f i e d a c c o r d i n g l y by t r e a t i n g v o l t a g e s on ground w i r e t o be z e r o .  This requires reducing  the impedance and admittance m a t r i c e s b e f o r e surge impedances can be calculated.  The  surge impedance v a l u e o b t a i n e d i n t h i s case i s lower than  t h a t o b t a i n e d by E q u a t i o n (2.29), as shown i n T a b l e  8.  2.4.  S i n g l e phase r e p r e s e n t a t i o n f o r c l o s e - b y s t r o k e s on circuited line  double  A f t e r the author has v e r i f i e d t h a t s i n g l e phase r e p r e s e n t a t i o n w i t h a p p r o p r i a t e c h o i c e o f l i n e parameters case  without ground w i r e  r  a  i s a c c u r a t e f o r a t h r e e phase  d o u b l e - c i r c u i t e d overhead  line  transmission line 23  of the MICA P r o j e c t o f the B.C.  Hydro and Power A u t h o r i t y  more d e t a i l e d t r a n s m i s s i o n system w i t h ground w i r e s .  was  used as a  37  surge  Impedances  ground w i r e  phase w i r e  A  547  ft  342 ft  B or C  545  ft  338 ft  D (A-B)/A*100%  -  318 ft  0.4%  1.2%  A = Exact method (2.38) w i t h Carson.':s C o r r e c t i o n terms f o r e a r t h r e t u r n s k i n e f f e c t , ground  wire  t r e a t e d as another phase. B = Exact method e q u a t i o n (2.38) without  Carson's  C o r r e c t i o n terms f o r e a r t h r e t u r n s k i n C = D a r v e n i z a approximate  equation  (2.29).  D = Exact method e q u a t i o n (2.38) w i t h  Carson's  C o r r e c t i o n terms f o r e a r t h r e t u r n s k i n ground w i r e t r e a t e d as e a r t h r e t u r n  Table 2.4:  effect.  effect,  component.  S e l f surge impedances f o r ground and phase w i r e s f o r a t y p i c a l 500 kV l i n e  ^ '  7  38  T h i s t r a n m i s s i o n system i s a d o u b l e - c i r c u i t e d 500 kV l i n e . tower c o n s i s t s of a t h r e e phase l i n e w i t h two and 2.5).  ground w i r e s  When the l i g h t n i n g s t r o k e h i t s e i t h e r one  or one of the phase c o n d u c t o r s , d i f f e r e n t because of d i f f e r e n t l i n e d e s i g n . i n T a b l e 2.5.  One  The  (See F i g u r e s 2 .4  of the ground w i r e s  l i n e parameters must be  the ground w i r e i s g r e a t e r than t h a t of the phase c o n d u c t o r .  In F i g u r e s 2.6  and 2.7,  i n the ground w i r e  one  F i g u r e 2.6a  short c i r c u i t t e s t  by  F i g u r e 2.6b  f o r the s i n g l e phase case and the o v e r a l l important  c h a r a c t e r t i s t i c s of m u l t i - p h a s e  the m u l t i -  I t a l s o shows c l e a r l y  d i f f e r e n t modal components on the r e s u l t i n g waveform.  the shows  propagation  representation i s successfully duplicated  S i m i l a r r e s u l t s are o b t a i n e d f o r the s h o r t - c i r c u i t t e s t , as shown  i n F i g u r e s 2.7a  and 2.7b.  o b t a i n e d from these two  One  can  successfully duplicated direct  recommended  strokes  or  observe  t h a t the c u r r e n t waveforms  d i f f e r e n t l i n e r e p r e s e n t a t i o n s agree v e r y w e l l .  S i m i l a r l y , the open and  of  case.  shows the r e s u l t o b t a i n e d by  phase s o l u t i o n method u s i n g modal a n a l y s i s .  here.  The wave  s i n g l e - p h a s e l i n e r e p r e s e n t a t i o n when l i g h t n i n g stroke,  h i t s the ground w i r e .  the r e s u l t  impedance of  can compare the l i g h t n i n g o v e r v o l t a g e  wave p r o p a g a t i o n c h a r a c t e r i s t i c s f o r the open and u s i n g m u l t i - and  chosen  c o r r e s p o n d i n g parameters are shown  can see from t h i s t a b l e t h a t the s e l f surge  p r o p a g a t i o n v e l o c i t y i s a l s o lower  Each  short c i r c u i t  t e s t r e s u l t s are a l s o  f o r surges on the phase conductor  as  in  case  b a c k f l a s h o v e r s ( See Fig.2.8&2.9) Thus, i t i s  t o use s i n g l e phase r e p r e s e n t a t i o n f o r d o u b l e - c i r c u i t e d l i n e  w i t h ground w i r e s  f o r studying close-by l i g h t n i n g stroke propagations.  23 u n i t 5-3/4x10" s h i e l d i n g ground tower i n s u l a t o r //wires extending strings / / 1.6km beyond substn  3 phase conduc tors  3 phase underground SF- c a b l e s 6-  tower  lightning arrester a t transformer ^jO^L-^ c a b l e j u n c t i o n J  .transformer t o be p r o t e c t e d Typical lightning arrester characteristics: nominal r a t i n g ( r e s e a l voltage), s w i t c h i n g sparkover Min 60 Hz sparkover l i g h t n i n g sparkover  Insulation  levels  60 Hz  BIL  F i g u r e 2.4:  420 950 568 985  kV kV kV kV  396 960 555 990  kV kVkV kV-  SF, c a b l e Transformer 6 800 kV 745 kV 1550 kV 1675 kV  Layout of SFg s u b s t a t i o n p r o t e c t i o n scheme showing one of the double c i r c u i t systems.  40  10 8  9  —  6  7  Conductors  Coupling  lightning struck  lightning  4 1  5  2  3  1-3, 6-8  phase w i r e s  4,5, 9,10  ground w i r e s  factor: ^phase surge ^phase surge  <j) - w i r e K 34  struck g - wire K  _ 43  3,4  •16  y induced ground w i r e V  3,3  ^phase surge 4,3 _ phase s u r g e 4,4 J  F i g u r e 2.5:  1  phase w i r e  induced phase w i r e  •06 V  ground w i r e  S i d e view of the MICA 10 <j> Systems.  Ground conductor  S e l f surge impedance  Z^|  Wave v e l o c i t y  v  Phase conductor  658  e  ft  304 ft  245 ™/\is  Line resistance  293  0 ^/m  Length  0  1609 m  where  Z  S ^ f self  1609  = 60 In  = /L  r  self  velocity =  m  /ys  ^/m  m  ^  P  77  self  self L  self  and L . , _. and P . , , are d i a g o n a l elements self 21 S  matrix  Table 2.4:  of  [L] and [ P ] .  L i n e parameters of ground and phase conductor f o r l o s s l e s s s i n g l e phase r e p r e s e n t a t i o n .  42  D i f f e r e n t modal a r r i v a l  S i n g l e phase a r r i v a l In 1<|> c a s e :  V  T 2 p 3  = (Coupling f a c t o r ) surge = Z  F i g u r e 2.6:  ' surge 4,4  3  4  V  T2G4  Open c i r c u i t t e s t on s i n g l e and m u l t i - p h a s e r e p r e s e n t a t i o n w i t h s t r o k e on ground w i r e .  times  4 3  i  D i f f e r e n t modal a r r i v a l times  Figure  2.7:  Short c i r c u i t t e s t on s i n g l e and m u l t i phase r e p r e s e n t a t i o n w i t h s t r o k e on ground w i r e .  F i g u r e 2.8:  Open c i r c u i t t e s t on s i n g l e and multi-phase representation with s t r o k e on phase c o n d u c t o r s .  45  Small d i f f e r e n c e i n d i f f e r e n t modal a r r i v a l  i T 2  t-0- \.  304«  TP3  T2P3  _3 1 0  P-u.)  4  1+ 2 p.u.  p3(  times  2  (S  Time T2P3  t} ii,.or  4  f  8  12-  16  (ys)  S i n g l e phase a r r i v a l  Figure  2.9:  Short c i r c u i t t e s t on s i n g l e and multi-phase r e p r e s e n t a t i o n with s t r o k e on phase c o n d u c t o r s .  46  CHAPTER 3:  LIGHTNING WAVE PROPAGATION IN S F GAS INSULATED UNDERGROUND TRANSMISSION CABLE SYSTEM. 6  1.  Introduction The world's f i r s t  345  kV was i n s t a l l e d  i n 1970.  underground o i l - f i l l e d l o s s e s , thermal  commercial SF, gas - i n s u l a t e d c a b l e r a t e d a t 6  c a b l e s w i t h r e s p e c t to c h a r g i n g  substation s i z e . switchgear  conventional  current,  dielectric  performance, v o l t a g e r a t i n g f l e x i b i l i t y and power h a n d l i n g  c a p a c i t y a r e well-known.  and  I t s i n h e r e n t advantages over  I t o f f e r s a d d i t i o n a l advantages o f reduced  T h i s compactness i n s i z e of S F ^ - i n s u l a t e d s u b s t a t i o n s  b r i n g s t h e equipment c l o s e r t o t h e p r o t e c t i v e l i g h t n i n g  a r r e s t e r l o c a t e d a t t h e overhead l i n e and underground c a b l e j u n c t i o n . T h i s i s o f . v i t a l - i m p o r t a n c e , e s p e c i a l l y when t h e r e i s no l i g h t n i n g a r r e s t e r a t the transformer  t e r m i n a l , as i n c e r t a i n  substation  design. 23 In the S F , - i n s u l a t e d c a b l e at t h e MICA Dam, which w i l l be used o as a t e s t example, each of t h e 3 phase c a b l e s c o n s i s t s o f two c o n c e n t r i c aluminum tubes (see F i g u r e 3.1a).  The i n n e r tube i s t h e conductor  the o u t e r grounded tube i s t h e sheath.  The t h r e e sheaths a r e s o l i d l y  bonded t o g e t h e r and grounded a t many l o c a t i o n s . encountered i n l i g h t n i n g surges,  c o r e and  At the high  frequencies  t h e sheath r e t u r n c u r r e n t w i l l be e q u a l  i n magnitude and 180° out o f phase w i t h the core conductor  current.  Whether the magnetic f i e l d e x t e r n a l t o t h e sheaths can be c o m p l e t e l y neglected  i n t h e frequency  range o f i n t e r e s t must be i n v e s t i g a t e d ,  however.  I f the magnetic f i e l d  i s n e g l i g i b l e , then t h e r e would be no  mutual i n d u c t i v e c o u p l i n g among t h e phases.  There i s no e l e c t r o s t a t i c  c a p a c i t i v e c o u p l i n g between phases as t h e s o l i d l y - g r o u n d e d sheaths a c t as e l e c t r o s t a t i c s h i e l d s p t o 1 Mhz. U  47  Skin depth o f A l ,  •t *  \-1/2  = (.TTfau)  8.1 =  cm  =1.0 cm a t 60Hz  Scale--= 1:2.54  = 3" = 7.62 cm  r  3.5" = 8.89 cm r„ = 9.75" = 24.765 cm r . = 10" = 25.4 cm 4 sheath thickness=.635 cm Permeability A l = y  r  u, = u0 0  -7 = 4TT .x 10 F i g u r e 3.1a:  I n d i v i d u a l cable  Permittivity  SF  &  = £  r  1  O v e r a l l cable  m  design  36TT  F i g u r e 3.1b:  H/  layout.  £  Q  =  x 10  e  9  Q  F/m  48  In o r d e r to i n v e s t i g a t e the sheath c a b l e s , and  c u r r e n t r e t u r n phenomenon of  thus to c l a r i f y the wave p r o p a g a t i o n  parameter must be c a l c u l a t e d a c c u r a t e l y . 24 parameters of m u l t i - c o r e c a b l e  c h a r a c t e r i s t i c s , the  SF,  b  cable  In t h i s r e s e a r c h work, c a b l e  25 '  i s not  investigated.  Such m u l t i - c o r e  c a b l e systems c e r t a i n l y have c o u p l i n g between phases at a l l f r e q u e n c i e s . Nevertheless, in different  c o u p l i n g between phases f o r the s i n g l e core c a b l e system f r e q u e n c i e s needs f u r t h e r i n v e s t i g a t i o n . 26  Commellini  27 and  Abledu  used f i n i t e elements technique  d i v i d e the main conductors i n t o s m a l l e r sub-conductors of shape. up  The  impedance m a t r i x  f o r t h e main conductors was  the sub-conductors i n the m a t r i x  e l i m i n a t i o n process.  t o sub-  cylindrical formed by  bundling  However, due  to  the t h i n t u b u l a r shape of the conductors i n v o l v e d i n the SF^ buses, l a r g e number of sub-conductors i s r e q u i r e d .  T h i s w i l l demand huge computer  s t o r a g e space and l o n g computer e x e c u t i o n time. 28 29 Sunde and P o l l a c z e k had d e r i v e d a n a l y t i c a l e x p r e s s i o n s the s e l f and mutual impedance of c a b l e s which are c o n s t r u c t e d underground or on ground s u r f a c e . K e l v i n f u n c t i o n s and terms.  Before  assumptions and process.  These a n a l y t i c a l e x p r e s s i o n s  contain  an i n f i n i t e i n t e g r a l known as the Carson's C o r r e c t i o n  r e s t r i c t i o n s were made to f a c i l i t a t e  With the r e c e n t p o p u l a r i t y  d i g i t a l computers, these  for  for  overhead,  the widespread a p p l i c a t i o n of d i g i t a l computers,  by s t r a i g h t forward  core  infinite  numerical  and  the  simplified  computation  i n c r e a s e d a p p l i c a t i o n of  i n t e g r a l s can be m o d i f i e d  computations without  and  significant  replaced sacrifice  accuracy. Wedepohl e t a l  30  and  Ametani  t a c k l e the a n a l y t i c a l e x p r e s s i o n  31  '  32  used d i f f e r e n t approaches to  i n the c a b l e parameter c a l c u l a t i o n .  However, b o t h approaches gave d i f f e r e n t  results  (20% from each o t h e r ) .  Bianchi  proposed t o c a l c u l a t e t h e e a r t h o r sea r e t u r n impedance by  approximating t h e r e t u r n medium as a tube These approximation r e s u l t s f e l l and  of  i n f i n i t e outside  radius.  somewhere between those o f Wedepohl et a l  Ametani. Because o f t h e i n c o n s i s t e n c y  i n the above f i n d i n g s , a d e t a i l e d  i n v e s t i g a t i o n f o r n u m e r i c a l c a l c u l a t i o n o f c a b l e parameters must be performed t o r e v e a l wave p r o p a g a t i o n c h a r a c t e r i s t i c s i n SF^ s i n g l e cables.  Current  r e t u r n c h a r a c t e r i s t i c s from core  must be i n v e s t i g a t e d t o c o n f i r m the c a b l e  2.  Formation o f s e r i e s impedance m a t r i x f o r SFfi The  system  through sheath a l s o  s i n g l e phase or m u l t i - p h a s e  for  s i n g l e core SF^ c a b l e  cables  system c o n f i g u r a t i o n i s shown i n F i g u r e  a 6 x 6 s e r i e s impedance m a t r i x Z, d e s c r i b i n g the c a b l e  c l dx  representation  involved.  Each phase c o n s i s t s o f two c o n d u c t o r s , core and s h e a t h .  d  system as f o l l o w s :  n  dV . si dx d V  Z  si l li  01  Z  Z  s22  Z  ml2  s21  c2 dx  Z  ml2  19  Z  ml2  Z  ml2  Z  ml3  Z  ml3  Z  ml2  Z  ml2  Z  ml3  Z  ml3  si l li  Z  sl2  3nl2  Z  ml2  sl2  z  s2 dx dV _ c3 dx dV _ s3 dx  "  ^ l "  ^1  ^2 •  dV Z  ml2  Z  ml2  Z  s21  Z  s22  Z  ml2  Z  Z  ml3  Z  ml3  Z  ml2  Z  ml2  Z  sll  Z  ml3  Z  ml2  Z  ml2  Z  _ ml3 Z  s21  3.  We can b u i l d up  V  z  core  ml2^  ^2  Z  sl2  ^3  Z  s22 _  . ^ 3 .  50  z  Z  s  Z,  ml2  "  ml3 ^1  Z  Z  ml2  Z  s  ml2  \l (3.1) ^2  Z  ml3'  Z  T  s where Z  s  is... ' s e l f submatrix  e q u a l because they It  is  assumed:  Z  ml2  on the d i a g o n a l . °  represent  ^3  s  A l l Z matrices are s  i d e n t i c a l cable configurations.  t h a t the mutual impedance between c o r e s , between  s h e a t h s and between c o r r e s p o n d i n g  c o r e s and sheaths a r e a l l e q u a l .  In  o t h e r words, a l l elements i n t h e sub-matrix Z , „ or Z. a r e ^assumed ml2 ml3 to,; be equall: :-(Bee S e c t i o n 6 f o r " f u r t h e r d i s c u s s ion) J  3.  •—  C a l c u l a t i o n o f s e l f and mutual e a r t h r e t u r n impedance f o r s i n g l e core c a b l e The a n a l y t i c a l e x p r e s s i o n s  f o r t h e s e l f and mutual e a r t h r e t u r n 29  impedance o f c a b l e s was f i r s t by  Wedepohl  d e r i v e d by Sunde and P o l l a c z e k ^  et a£? F i r s t l y , t h e Maxwell's e l e c t r o m a g n e t i c  be s o l v e d a f t e r n e g l e c t i n g end e f f e c t s a s :  and then  equation  can  51  V x E = -ja>y H  (3.2)  0  V x H = J + |2. = j ( i + 3t  (3.3)  a  = J , as displacement c u r r e n t be n e g l e c t e d .  Taking  the c u r l  of e q u a t i o n  (3.2) and s u b s t i t u t i n g i n t o (3 3) g i v e  V x V x E = -jwy  or  V(VE)  0  V x H  - V E = -juy (J) 2  Q  Assuming c a b l e s e p a r a t i o n >> c a b l e r a d i u s , we V E =  jtoy  2  Assuming c a b l e s a r e p a r a l l e l v o l t a g e and c u r r e n t  0  have  a(E +p i 6(x) <5(y + h))  t o ground s u r f a c e and a t t e n u a t i o n of  i s n e g l i g i b l e over d i s t a n c e s comparable to c a b l e 2 3 E  we have  , can  3x  2 3 E,  n  2  2 3 E„ —j3x  separation,  + —T-  3y  +  =  2  2 3 E Y 3y  » y > o.  0  = m E 2  2  +p m  2  (-> 3  i 6(x) 6(y + h) , y < 0.  where  m  4  (3.5)  = J j cjy P  : ;  1  . . .  P •• = e a r t h r e s x s t i v x t y = — oj = a n g u l a r  frequency  y = permeability 6 = Difac function E^,E Imposing the r e s t r i c t i o n f o r E^ and  2  = electric  field  above and below ground.  of c o n t i n u i t y of e l e c t r i c  field  as boundary c o n d i t i o n s , we can o b t a i n a g e n e r a l  at y = 0  expression  52  y  7TT  Figure  3.2:  Conductor c o n f i g u r a t i o n s  for earth return  formula.  53  f o r the underground e l e c t r i c for  .  We can then o b t a i n t h e e x p r e s s i o n  t h e mutual e a r t h - r e t u r n impedance  E ( x , h ) by t h e c u r r e n t 2  field  i (O.h^)  2  t  8  o  o f underground c a b l e s by d i v i d i n g  e t  / 2" 2*  00  exp(-£/ a +m  Z. . = ij 2IT  1  '/2 2~\  00  ) j ax ^  +  &  j exp;(-l//a +m  0/2.2 2/a +m  ' n~~2 + Va +m  1  a  )  exp( - £ V a + m ) 2  2v4 +m 2  exp,(= v^/a -rm : 2  2^.00  = A Z ij 4 J  J  where  1  a I  + 4 ^ 1  277  2  +/a +ra 2  lax  da +  2 77  (K (mDj n  s  (T  1  da -  2  da  (3.6)  2  -  (K (mD ))  0: 2' 0  0  2  fe-JuDj  ^ o ^ l  - K„(mD„)) y  = K e l v i n f u n c t i o n of o r d e r  CT  2  zero  m = y ^ ^ " j x = h o r i z o n t a l s e p a r a t i o n between c a b l e s D, = /x  D  /2  2  2  =  2  = depths o f b u r i a l o f c a b l e s  h^,h  a  /x  +(^4-^)  = l = |h + h I 1  AZ.. = Carson's C o r r e c t i o n term, i d e n t i c a l t o t h a t f o r overhead l i n e s  case.  (3.7)  The above formula i s a l s o a p p l i c a b l e to s e l f e a r t h r e t u r n components.  In such c a s e s , the terms f o r  D D  1  2  and  impedance  can be r e - d e f i n e d as  r  =  = 2h  (3.7a) •  where r i s r a d i u s , h = depth o f b u r i e d  cable.  However, the above f o r m u l a i s u n s u i t a b l e f o r s t r a i g h t forward c a l c u l a t i o n s . B e f o r e d i g i t a l computers  a r e w i d e l y used, approximate r e s u l t s were o b t a i n e d 28  only a f t e r c e r t a i n l i m i t i n g conditions Wedepohl e t a l used Cauchy's  were accommodated.  Then,  i n t e g r a t i o n f o r t h e Carson's c o r r e c t i o n  terms and d e r i v e d approximate f o r m u l a f o r t h e e q u a t i o n g i v e n i n e q u a t i o n (3.7) as mD „ Z. . = ^ { £n(Y-±-) + i - f mA } ij • 2 ir 2 2 i for  and  |mD  (3.8)  | < 0.25  Y = Euler;'s c o n s t a n t = 0.5772157.  However, the above formula g i v e s r e s u l t s which a r e about 20% h i g h e r when compared w i t h the d i r e c t n u m e r i c a l computation u s i n g t h e o r i g i n a l equation  as i n Ametarii's c a s e .  The i n f i n i t e  i n t e g r a l f o r t h e underground  c a b l e i s the same as the Carson's c o r r e c t i o n terms f o r overhead  lines.  We can d e f i n e a new parameter a as  a = / ^ D  =  4TT  /5 10  4  D/|-  (3.9)  55  where  D and  p are i n MKS  units  2h. f o r s e l f ^ ~  earth return  impedance  f o r mutual e a r t h r e t u r n  impedance  T h i s c o r r e c t i o n term i n t e g r a l can be r e p r e s e n t e d by the f o l l o w i n g . .„ . . 3 4 i n f i n i t e converging s e r i e s : 1.  For a < 5  AR' = 4o)-10~ {^4  o  -b^a* cost)) +b^[ ( c 2 ~ l n a ) a  2  2 cos2<{)+(j)a sin2cj)]  3 +b^a cos3<}> -d^a cos4cf> 4  -b^a^cos5<j>  6 6 +b^ [ ( c ^ - l n a ) a cos6<j)+cba sin6c))] +b^a^cos7<j> g - d a cos8<f> Q  o  - ...} AX' = 4to-10 {|<0.6159315-lna) 4  +b^a*cos<j> 2 -d^a cos2cf> 3 +b^a cos3<j> 4 4 - b ^ [ ( c ^ - l n a ) a cos4cf>+(j>a sin4cf>] +b^a^cos5cj) -d^a^cos6tj) +b^a  cos7<j>  8 8 - b [ ( c - l n a ) a cos8<J>+c(>a sin8c()] Q  o  Q  o  + ...}  56  N o t i c e t h a t each 4 s u c c e s s i v e terms form a r e p e t i t i v e p a t t e r n .  The  co-  e f f i c i e n t s b., c. and d. are o b t a i n e d from the r e c u r s i v e f o r m u l a s : l  1 1  ^  z^foN  b. = b. „ . l i-2 i ( i + 2 )  /2 b^ = — - r f o r odd  w i t h the s t a r t i n g v a l u e -\ 1  ^•b„ = 2 c  l  = c. „ + + -rrr w i t h the s t a r t i n g v a l u e i-2 l i+2 -  A  subscripts,  ID  f°  r  even s u b s c r i p t s ,  c„ = 1.3659315, 2  U  7 1  d. = j- • b., l 4 l w i t h s i g n = ±1 changing  a f t e r each 4 s u c c e s s i v e terms ( s i g n = ±1 f o r i  1,2,3,4; s i g n = -1 f o r i = 5,6,7,8 e t c . ) . 2.  For a > 5  , _ ' cos((> - (  /2 cos2<)) cos3<}> 2 3 5  3cos5<j) "  45cos7(j) 7 )  4a)-10  4  /I Q3.ll)  AX  1  - (  C O S 1  ^  _  a  It  should be noted  cos  3<l> 3 a  +  3cos5<{> 5 a  45cos7(}> ^ 7 a  +  #  -4  4ai* 10 >=v2  t h a t the c o r r e c t i o n terms w i l l become zero when  the parameter a i s v e r y b i g , i e . when f r e q u e n c y or c a b l e d i s t a n c e from g.round  i s very l a r g e or when e a r t h r e s i s t i v i t y  i s very s m a l l .  The K e l v i n f u n c t i o n s can a l s o be c a l c u l a t e d by another converging s e r i e s as  infinite  36 can be o b t a i n e d from a v a i l a b e source. , ''-It can  also  35 be c a l c u l a t e d by a s p e c i a l s u b r o u t i n e CBESK Computing  a v a i l a b l e from t h e  UBC  Centre.  A f t e r e s t a b l i s h i n g the n u m e r i c a l formula  for  cable earth return  impedance, the d i s c r e p a n c i e s between the r e s u l t s of Wedepohl e t a l and  those of Ametani can then be The •can  author has  clarified.  confirmed  that accurate  c a b l e e a r t h r e t u r n impedance  be c a l c u l a t e d by u s i n g d i r e c t computation of i n f i n i t e s e r i e s  sub-  40 s t i t u t i o n f o r i n f i n i t e i n t e g r a l and  Kelvin function.  impedance from Ametanis computation i s a c c e p t a b l e  though d i f f e r e n t  s e r i e s i s used f o r the Carson's c o r r e c t i o n terms. obtained  a t l e a s t to 4 s i g n i f i c a n t  (See F i g u r e 3.2).  The  approximate formula  author has  infinite  Identical results  f i g u r e s f o r f r e q u e n c i e s up  o t h e r hand, g i v e s r e s u l t s about 20% The  Cable mutual  g i v e n by e q u a t i o n  are  to 100 (3.8)  k  on  Hz the  c o n s i s t e n t l y higher.  a l s o confirmed  t h a t the e a r t h r e t u r n impedance f o r  underground c a b l e can be approximated by the e q u i v a l e n t e a r t h r e t u r n impedance f o r overhead l i n e s .  The  expression  f o r e a r t h r e t u r n impedance  f o r overhead l i n e i s Z.. ij  =  to 2TT  D -==- + AZ.. , DJ^ IJ '  .„ (3.12)  f o r mutual e a r t h r e t u r n impedance  and  Z. . =  to 2 It  xj  2h — GMR  + AZ. . ,  (3.13)  xj  f o r s e l f e a r t h r e t u r n impedance  where  i s d i s t a n c e between i * " *  1  and  i s d i s t a n c e between i * " *  1  and j '  • h i s height GMR AZ  image of j " ^  conductor;  1  t  1  conductor;  of conductor above ground;  i s geometric mean r a d i u s =  r a d i u s of conductor a t h i g h f r  i s Carson's c o r r e c t i o n term.  I d e n t i c a l r e s u l t s up to 3 or 4 f i g u r e s are o b t a i n e d r e s i s t i v i t y of 1 t o 100  Q-m  up t o the frequency  of 100  for earth  K Hz.  The  58  c o n s i s t e n c y between these two r e s u l t s i s due to the f a c t t h a t t h e K e l v i n functions  K (/j" x) = k e r ( x ) + j kei(x).;  can be e v a l u a t e d by the f o l l o w i n g convev^ing s e r i e s  36  f o r 0 < x < 8:  ker x = — In ( i r ) b e r z + J x b e i x—.57721 566  1  -59.05819 744(jf/8)*+171.36272 133(x/8)  8 :  -60.60977 451(z/8) +5.65539 121 (x/8)' 12  8  -.19636 347(x/8) +.00309 699(:r/8) 20  24  -.00002 458(2-/8) +« 28  H<ixio-  (3.14)  8  kei*=-ln($x)beia;-fir ber a,+6.76454 936(z/8)* -142.91827 687(i/8) +124.23569 Q50(z/8y° 6  -21.30060 904(ar/8) +l. 17509 064(r/8) 14  18  -.02695 875(ar/8) +.00029 532(x/8) +« 22  29  _(3.15)  |€|<3X10-»  where ber x=1 -64(as/8) +113.77777 774(*/8) 4  8  -32.36345 652(z/8) +2.64191 397(a;/8) 12  16  -.08349 609(*/8) +.00122 M  552(x/8)  24  -.00000 90l(x/8) +e 28  |«|<ixio-»  bei x=16(a;/8) -113.77777 2  774(x/8)*  + 72.81777 742(a/8) ~ 10.56765 779(x/8) l0  u  + .52185 615(z/8) -.01l03 667(ay8) l8  22  +.00011 346(z/8) +e 29  M<8X10-»  (3.16)  F i g u r e 3.2:  Mutual impedance between outermost by d i f f e r e n t computation methods.  cables  60  and  x = ^f-  D,  (3.18)  u = 4TT x 10 ^ oi =  ^m  frequency  D = d i s t a n c e Di  o  r  L  P  = earth  D  2  resistivity  For f r e q u e n c i e s up t o about 1 M Hz and e a r t h r e s i s t i v i t y  o f about  100 ft-m,and c a b l e s e p a r a t i o n o r c a b l e depth o f about 1 m, the term x i s r e l a t i v e l y s m a l l as  x -> 0  Then, one can r e w r i t e e q u a t i o n s  (3.14) t o (3.17) f o r x  0 as  ber = 1 bei • •  K (/fx) Q  = 0 = ker x + j k e i x = - £n |- x - 0.57721 - j J  (3.19)  Thus, f o r the e a r t h r e t u r n impedance as shown i n e q u a t i o n ( 7 ) , we have  Z. . = AZ. . + ™ xj xj 2 IT  (K (mD.) - K (mD.)) o 1 o 2 mD  n  - « i f « - «• -r 4Z  +  - jj) -  mD ( - in -f - - .57721 - jJ-))  = AZ . + ^  y  1  J  Carson's c o r r e c t i o n term  2lT  • j,  in ^  ,  x * 0  °1  self-term  which i s the same as i n equations  (3.12) and  (3.13).  61  The n u m e r i c a l  r e s u l t s f o r the s e l f - t e r m component of the  self  and mutual e a r t h r e t u r n impedance f o r the underground c a b l e o b t a i n e d the d i f f e r e n t can be  formulae developed e a r l i e r  seen from T a b l e  3.1,  are shown i n Table  the r e s u l t s o b t a i n e d by these  formulae are very c o n s i s t e n t .  The  3.1.  different  f o r f r e q u e n c i e s up  1 M Hz.  Because of the observed c o n s i s t e n c i e s , the overhead  formula  approximation i s t h e r e f o r e recommended f o r underground  4.  to 1 M Hz  C a l c u l a t i o n of s e l f  As  f i n a l r e s u l t s f o r mutual impedance  from these methods are a l s o shown i n F i g u r e 3.3  f o r a l l f r e q u e n c i e s up  by  and  to  line cable  e a r t h r e s i s t i v i t i e s above 1 ft-m.  impedance m a t r i x  f o r s i n g l e core  cable  A f t e r the s e l f and mutual impedance f o r e a r t h r e t u r n o f under^. ground cables,.is o b t a i n e d , be c a l c u l a t e d and  the o b t a i n e d  then be transformed submatrix Z  s  the s e l f  impedance of i n d i v i d u a l c a b l e s  can  r e s u l t s f o r d i f f e r e n t c u r r e n t loops  can  to the r e q u i r e d form f o r the impedance  as shown i n e q u a t i o n  At f i r s t ,  one  (3.1).  can c o n s i d e r the c u r r e n t i n each of the  c a b l e s flow i n two a d j a c e n t  loops as shown i n F i g u r e 3.4.  formed by the c u r r e n t f l o w i n g through the c o r e and the o u t s i d e sheath. sheath and  be d e s c r i b e d by e q u a t i o n  (3.21) as  Z  dx  l l  Z  12  1  1  4  where Z  2 _ 21 Z  = Z  Loop 1 i s  r e t u r n i n g through  These two  d v l  dx  individual  Loop 2 i s formed by the c u r r e n t f l o w i n g through the  r e t u r n i n g through the o u t s i d e e a r t h .  d v  diagonal  by  Z  22  symmetry.  _ 2_ i  loops  can  62  S e p a r a t i o n i n meter D  (1,3)  (1,2)  (self)  l  .889  1.778  .254  D  underground  overhead  2  2.189  .901+j.OOO  2.676  2.0  (60Hz)  .901  .895-J.020 (100Hz.)  .901  .408+j.OOO  (60Hz)  .408  .403-J.018  (100Hz)  .408  2.064+j.OOO  (60Hz)  2.064  2.057-J.022 (100Hz)  2.064  where f = 60 Hz,  m  =/  j  6  °  X  2  \ l  ^  Q  X  1 0  " -= 7  i TT /icoy / j l O O x 2TT x 4TT X 10 ^ f = 100 Hz, m = / ^ = / JOO J  underground c a b l e  /J  .0022  rr -089 / j A  Q  n  (exact)  Z  ij  =  Trf  (  V  m  V  '  K  0  (  m  D  2»  +  A  Z  ij  overhead c a b l e (approximation t o above)  hi T a b l e 3.1:  " ^ *> 57  +  * 13 Z  •  - -  » * 0  Mutual and s e l f e a r t h r e t u r n impedance terms as g i v e n by underground and overhead c a b l e f o r m u l a .  F i g u r e 3.3:  Approximations of mutual impedances between underground c a b l e s by Carson's formulae.  F i g u r e 3.4:  Current loops i n s i d e SF^ c a b l e f o r s e l f impedance c a l c u l a t i o n .  65  The m a t r i x elements o f e q u a t i o n the i n d i v i d u a l c u r r e n t l o o p  (3.21) can be o b t a i n e d by c o n s i d e r i n g  components making up the c o r r e s p o n d i n g  loops  1 and 2 as  Z  Z  11  22  z  12  = Z  core-outside  + Z + Z core/sheath i n s u l a t i o n sheath-inside  = Z + Z sheath-outside earth-inside  = z  21  (N.B. Z ^  (3.23)  = —z  sheath-mutual (minus s i g n s i n c e i different  (3.22)  and ±  in  (3.24)  direction).  i s n e g l i g i b l e when sheath t h i c k n e s s »  s k i n depth)  where the i n d i v i d u a l elements a r e (Zl)  Z  i n t e r n a l impedance o f c o r e w i t h r e t u r n  core-outside  through o u t s i d e ( s h e a t h ) . (Z2)  Z  core/sheath  insulation  impedance of SF, i n s u l a t i o n due to the o time v a r y i n g magnetic f i e l d .  (Z3)  Z  sheath-inside  i n t e r n a l impedance o f sheath w i t h r e t u r n through i n s i d e  (Z4)  Z  sheath-outside  (core).  i n t e r n a l impedance o f sheath w i t h r e t u r n through o u t s i d e ( e a r t h ) .  (Z5)  Z  earth-inside  self  e a r t h r e t u r n impedance, t h i s can  be c a l c u l a t e d by equations or  (3.7) & (3.7a),  can a l s o be o b t a i n e d by e q u a t i o n  w i t h the a p p r o x i m a t i o n  of  (3.25)  infinite  33 outside radius  (Z6)  Z  sheath-mutual  = mutual impedance of t u b u l a r sheath between loop 1 i n i n n e r s u r f a c e and loop 2 i n outer s u r f a c e of sheath.  66  The  individual self  and mutual impedance terms can be again  obtained  by s o l v i n g the Maxwell's e q u a t i o n s f o r the c o a x i a l conductors as s i m i l a r t o e q u a t i o n s "£3.2)&(3 .3) Schkelkunoff  J  37  and  They are a f u n c t i o n of frequency as d e r i v e d  #  Sunde  28  as  . ., tube-inside  =  tube-outside  = 2~-  tube-mutual  n  Q  L  = angular  K  (3.26)  Q  - I (mq)  K  (mr)  permeability = P ^ Q J  q  o u t s i d e r a d i u s of t u b u l a r conductor  r  i n s i d e r a d i u s of t u b u l a r p = d.c.  p  i  0'' 1 K  (3.28)  frequency = 2i:f  y  m r  (3.25)  (3.27)  1  V  + K„(mq) I.,(mr)) 0 1  ( I ( m r ) K (mq) + K ( m r ) I ^ m q ) )  p = I ( m r ) ^(mq)  where  and  (I (mq) K, (mr) 0 1  2iTqp  2Trqrp  with  Bessel  functions  Kelvin  functions  y  = r  1 for Al  conductor  resistivity  A f t e r o b t a i n i n g the i n d i v i d u a l terms of the loop e q u a t i o n as shown i n e q u a t i o n  (3.21),  we  can  then o b t a i n the d i a g o n a l  elements by a p p l y i n g the f o l l o w i n g c i r c u i t  V  1  V  = V = V  2 1  1  =  1  i» = i 2  c  by  - v  s  matrix  sub-matrix  conditions: (3.30)  s (3.31)  c  c  + i  s  (3.32)  The i n d i v i d u a l loop equations dV  of equation  (3 2 1 ) then becomes  dV  -T dx" 9  + -J^ dx  = ( 11 n + Z  z  ic+  i12 , )  Z  i o  12  (3.33)  s  1  dV and  —  dx  Adding equation  =  (Z  (3.34)  Z,_) i + Z„„ i 22 c 22 s  +  12  (3.33) to (3.34) g i v e s  dV ( Z  dx  11  +  2 Z  12  +  Z  22>  \  +  ( Z  12  +  Z  2 2 ) is  '  (  3  "  3  5  )  Thus, we can r e w r i t e t h e s e l f sub-matrix Z as ' s dV  c dx  Z  1 1  +  2  Z  1 2  +  Z  2 2  Z  12  +  Z  22 (3.36)  dV" s dx  5.  Z  12  +  Z  22  J  /ffheath c u r r e n t refragn c h a r a c t e r i s t i c s for, u s u a l  22  earth  As c u r r e n t flows a l o n g the core o f the b u r i e d SF^ bus, a r e t u r n path i s formed on i t s own sheath s o i l and a d j a c e n t i t s own sheath skin effect  sheaths.  and p o s s i b l y a l s o on the s u r r o u n d i n g  Whether  a l l the c u r r e n t s w i l l  depends s o l e l y on the f r e q u e n c i e s  involved.  r e t u r n through Due t o t h e  i n sheath m a t e r i a l (Aluminum), a l l core c u r r e n t w i l l  through i t s own sheath  return  f o r f r e q u e n c i e s above 1 k Hz.  In r e a l i t y , the SF^ c a b l e i s l a i d on the ground s u r f a c e  (e.g. i n s i d e  the l e a d s h a f t ) o r i s c o n s t r u c t e d above ground and grounded a t c e r t a i n intervals  ( e . g . i n s i d e the s u b s t a t i o n ) .  more c u r r e n t r e t u r n i n g through the sheath buried'cable  case.  One can thus  t h e c a b l e b u r i e d underground.  T h i s c a b l e l o c a t i o n even  favor  than the ground as compared t o  i n v e s t i g a t e the l i m i t i n g case with  T h i s c a b l e l o c a t i o n w i l l favour  least  68  core c u r r e n t r e t u r n i n g through i t s own  sheath.  In o r d e r to i n v e s t i g a t e the sheath and  t h e r e f o r e the  mutual  s e r i e s impedance m a t r i x . are e i t h e r i n c l u d e d or a.  current return  c o u p l i n g between c a b l e s , one One  can  c o n s i d e r cases  characteristics has  to use  the  i n which a d j a c e n t  sheaths  excluded.  Sheath c u r r e n t r e t u r n c h a r a c t e r i s t i c s f o r s i n g l e c a b l e  For t h i s case, one  o n l y has  of the s e r i e s impedance m a t r i x  to c o n s i d e r the s e l f d i a g o n a l  system submatrix  as  dV sll  J  dx  J  sl2 (3.37)  dV dx  J  sl2  J  s22  S i n c e the sheaths of the t h r e e i n d i v i d u a l SF^ grounded at short j o i n t  a l l p r a c t i c a l purposes.  J  or  v o l t a g e s can be  Then E q u a t i o n  i  sl2 Z  s!2  's22  I  c  c  +  Z  considered  (3.37) becomes  At h i g h f r e q u e n c i e s as sheath mutual impedance i s n e g l i g i b l e ^ 3 3 ) when sheath t h i c k n e s s g r e a t e r \ t h a n s k i n depth( See Appendix A . ) .  -  sheaths,  the c u r r e n t  c h a r a c t e r i s t i c s of the SF,, c a b l e through i t s own o can be c a l c u l a t e d as i n Equation  to be z e r o f o r  s22  Thus, n e g l e c t i n g the o t h e r two  shown i n F i g u r e 3.4 ..  solidly  i n t e r v a l s or l a i d on e a r t h s u r f a c e , or b u r i e d  i n s i d e .the e a r t h , the sheath  0  c a b l e s are  (3.38).  The  sheath  obtained  return  from the results  core are  For t h i s c a s e , e s s e n t i a l l y a l l the c u r r e n t  the core w i l l r e t u r n through i t s own  sheath  above the frequency  through  of 10  Hz.  69  z  0  sheath-mutual  '''sheath earth  """core _  "?"sheath core  s i n g l e - c a b l e system t h r e e - c a b l e system: - c u r r e n t i n one c o r e • ' -current i n three cores  ®-  i  B  l  /i  c  l  ® "  i  s2  / l  c2  ® "  1  s3  / l  c3  p = lOOft-m depth = 0 or .254m  log  frequency 1—  10  F i g u r e 3.4:  100  lk  R a t i o of c o r e c u r r e n t r e t u r n through own f o r s i n g l e - a n d t h r e e - c a b l e system.  10k  sheath  (Hz)  70  In o t h e r words, the sheath frequency  o f 10 Hz.  a c t s as a p e r f e c t magnetic s h i e l d above t h e  Because o f t h i s c o n s i d e r a t i o n , a l l SF^ c a b l e s a r e  decoupled from one another and can be r e p r e s e n t e d systems.  I t i s a l s o found  as 3 s i n g l e phase  that a change i n depth (1 m t o .254 m) o f  c a b l e does n o t change t h e c u r r e n t r e t u r n c h a r a c t e r i s t i c s n o t i c e a b l y .  b.  Sheath r e t u r n c h a r a c t e r i s t i c s f o r 3-cable system  S i n c e a l l t h e 3 sheaths o f t h e SF^ bus a r e s o l i d l y grounded, the current w i l l (< 60 H z ) .  r e t u r n through a l l t h e t h r e e sheaths a t lower At h i g h e r  frequencies  f r e q u e n c i e s , however, a l l the core c u r r e n t  r e t u r n through i t s own sheath  will  because o f t h e s k i n e f f e c t on t h e sheath.  Here, a g a i n , one can conclude t h a t the t h r e e SF^ c a b l e s a r e decoupled from one another. For t h i s case o f 3-cable v o l t a g e s t o be z e r o . and  system, one can a l s o c o n s i d e r t h e sheath  One can s u b s t i t u t e t h i s c o n d i t i o n i n t o e q u a t i o n  (3.1)  obtain dV  " cl  dx 0 d V  c2 dx 0  d V  c3 dx 0  (N.B.  -  5sii  Z  sl2  Z  ml2  Z  ml2  Z  ml3  Z  ml3  ^1  Z  sl2  Z  s22  Z  ml2  Z  ml2  Z  ml3  Z  ml3  ^1  Z  ml2  Z  ml2  Z  sll  Z  sl2  Z  ml2  Z  ml2  \2  ml2  Z  ml2  Z  sl2  Z  s22  Z  ml2  Z  ml2  i  'ml3  Z  ml3  Z  ml2  Z  ml2  Z  sl2  c3  ml3  Z  ml3  Z  ml2  Z  ml2  Z  s22  s3  Z  Z  Z  Z',„ = Z as symmetrical ml2 mz 3 i n F i g u r e ,3.1b)  Z  s lnl  Z  sl2  s2  arrangement o f c a b l e s as  (3.39)  E q u a t i n g the zero sheath v o l t a g e s f o r t h e 3 c a b l e s , we  0 = Z  . + Z „ i , + Z. . . ( i + i ) + Z (1 + i ) cl s22 si ml 2 c2 s2 ml 3 c3 s3  i  c 1 0  have  sl2  0 = Z (i.-. + i ) + Z ml 2 cl si s 1 0  l 2  i  . + Z i s22 s2  + Z (1 . + i ) ml2 c3 s3  n  0  (3.41)  1 0  c2  0 = Z , ( i - + i ) + Z. , ( i „ + i ) + Z _ i ml3 cl si ml 2 c2 s2 s l 2 0  (3.40)  _ + Z „ i _ c3 s22 s3  0  0  ,~ . (3.42) / 0  I f we assume phase B i s e n e r g i z e d , i . e . , we assume ±  c l  = i  c  (3.43)  3  ^1 = S 3  i  Then, s u b s t i t u t e  0 = Z  i  c 0 0  S22  si  t  l  = i  c  = 0  3  equations  + Z ml2  ( 3  -  4 4 )  (3.45)  (3.43) t o (3.45) i n t o  i + Z s2 ml3  i si  + Z m!2  (3.40), we o b t a i n  i c2  = (Z. , _ + Z ) i + Z. i + Z i ml 3 s22 si ml2 c2 ml2 s2 n  Z  or  nil3  +  Z  s22  X  s l , s2 +  ml2  1  Also s u b s t i t u t e 0  "  2 Z  ml2  \l  ! ! i ^ . i 5 l Z  sl2  X  c2  c2  1  equations +  Z  s 2 2 \2  +  , = -1  1  sl2  (3.43) tO (3.45)  ^ 1  (3.46  c2  Z  +  !522 Z  / 0  =  c2  into  (3.41), we o b t a i n  sl2  _  ±  (3.47)  72  By s o l v i n g equations  ml3  Z  +  Z  (3.46) and (3.47), we get  s22  -1  ^ml2 2Z  ml2  -1  sl2  J  (Z ... + Z ) Z - 2Z ml3 s22 sl2 ml2 0 0  2Z  Z  ml3  +  J  Z  ml2' mi2 Z  Z B  22  ( Z  ml3  +  "  Z s  J  22  ml2  (3.48)  )  s22  ml2  2Z  ai2  'sl2  J  Z  s22  sl2  -1  -1  J  s22  J  sl2  -Z 2Z  Z  ml3  +  Z  sl2  m!2  Z  Z  ml2  ml2  +  Z  Z  s22  s22 ml3 ( Z  J  ml2 +  Z  (3.49)  s22  )  s22  ml2 2Z J  nil2  J  s22  sl2  J  sl2  The r a t i o of c u r r e n t s i n sheath  to c o r e of phase B i s c a l c u l a t e d  and p l o t t e d as a f u n c t i o n o f frequency i n F i g u r e 3 . 4 . The at f r e q u e n c i e s above 1 kHz  result  shows t h a t  as i n l i g h t n i n g s u r g e s , a l l c u r r e n t through the  core w i l l r e t u r n through i t s own e a r t h r e t u r n c u r r e n t component  sheath.  Thus, one can conclude t h a t the  i s not important and t h e r e f o r e t h e mutual  73  c o u p l i n g between c a b l e s can be i g n o r e d .  c.  Sheath c u r r e n t r e t u r n c h a r a c t e r i s t i c s f o r 3 c a b l e system w i t h c u r r e n t i n a l l 3 cores  For a c a s e when c u r r e n t s flox^ the  three  SFg  accordingly.  cables, the return  in  a l l the  current  T h i s suggests that such  cores o f  through sheath w i l l  change  s i t u a t i o n s must a l s o be i n v e s t i g a t e d  to deduce the mutual c o u p l i n g e f f e c t among c a b l e s . as d e r i v e d i n E q u a t i o n s  three  (3.40) t o (3.42),  Using  t h e same e q u a t i o n s  one can now put i n c u r r e n t s i n the  3 cores by assuming  Z  Z  Z  S22  \l  ml2 \l  ml3 S i  equations  +  2  -  i . o IR  (3.50)  c  i  c  3  = 1.0 /120°  (3.51)  i  and  Rewriting  i  Z  +  Z  +  Z  ml2 ^ 2  +  s22 S 2  ml2 S  = 1.0  •  /-120' '  (3.52)  (3.40) t o (3.42) as  Z  ml3 ^ 3  V ?  +  +  cl  Z  2  ' s l 2 \l  =  Z  " " ,12 S i  ^3  s22 S  " ml2 ^ 2  Z  Z  = " ml3 S i Z  3  " s l 2 c2 Z  ±  " ml2 S Z  " ml3 ^ 3 = h  " ml2 S 3 Z  ~ sl2 S Z  2  <-  Z  =  = 3  A  A  2  3  3  ( 3  ( 3  "  '  53)  5 4 )  5 5 )  D e f i n i n g the determinant T as  Z  s22  Z  ml2  Z  ml3  Z  ml2  Z  s22  Z  ml2  Z  ml3  Z  ml2  Z  s22  Z  s2 2 3  +  2 Z  ml2  ' ml3 " s 2 2 m l 3 Z  Z  ( Z  +  2 Z  m  12>  (3.56)  74  We  can then o b t a i n the c u r r e n t through the t h r e e i n d i v i d u a l sheaths as'  i  2 = l s22 t A  s  l  2  Z  A  3 ml2 Z  +  2 A  2  Z  12 ml3 " 3 s22 ml3 ~ 2 ml2 22 " Z  m  A  Z  Z  A  Z  Z  A  s  l ml2 Z  T (3.57) i  2 = 2 s22 A  2  Z  +  A  l ml2 Z  Z  ml3  +  A  2 3 ml2 ml3 ~ 2 ml3 ~ 3 ml2 s22 " Z  Z  A  Z  A  Z  Z  A  l ml2 s22 Z  Z  T (3.58)  1  S3  2 " 3 s22 A  Z  +  A  2 rql2 ral3 Z  Z  +  A  2 l r a l 2 " l m l 3 s22 " 2 ml2 s22 " 3 ml2 Z  A  Z  Z  A  Z  Z  A  2  Z  T (3.59)  After substituting equations  the c o n d i t i o n s , f o r the 3 phase c u r r e n t s from  (3.50) to (3.52) i n t o e q u a t i o n s  (3.57) t o (3.59), one can o b t a i n  the r e t u r n c u r r e n t s through a l l i n d i v i d u a l sheaths. sheath c u r r e n t s are a l s o shown i n F i g u r e 3 . 4 . i t  The magnitudes o f the  i s a g a i n confirmed  here  t h a t at frequency above 60 Hz, a l l c u r r e n t f l o w i n g from core w i l l r e t u r n through i t s own cores.  sheath.  Each core i s c o m p l e t e l y s h i e l d e d from t h e a d j a c e n t  Thus, the t h r e e SF^ buses are c o m p l e t e l y decoupled  from one  another  and should t h e r e f o r e be r e p r e s e n t e d by s i n g l e phases as i n the case of l i g h t n i n g overvoltage propagation.  6.  Sheath c u r r e n t r e t u r n c h a r a c t e r i s t i c s f o r s u b s t a t i o n e a r t h w i t h grounding g r i d network In r e a l i t y  i n the s u b s t a t i o n , the c a b l e sheaths are grounded  inside  the s u b s t a t i o n w i t h a grounding network g r i d c o n s i s t i n g of copper  bars  which a r e connected  copper  b a r s s e r v e to reduce substation.  a c r o s s the whole s u b s t a t i o n . s i g n i f i c a n t l y the  T h i s suggests  inside  These grounding  e a r t h r e s i s t i v i t y of the  that the sheath c u r r e n t r e t u r n c h a r a c t e r i s t i c s  of t h i s reduced e a r t h r e s i s t i v i t y  should a l s o be i n v e s t i g a t e d as a r e d u c t i o n  75  i n e a r t h r e s i s t i v i t y w i l l f a v o r more c u r r e n t r e t u r n i n g through the The  r e s u l t f o r the sheath  resistivity  r e t u r n c u r r e n t as a f u n c t i o n o f  i s shown i n F i g u r e 3.5.  T h i s f i g u r e shows that the  r e t u r n c u r r e n t i n c r e a s e s as the e a r t h r e s i s t i v i t y w i t h the manufacturers and u t i l i t y  companies of SFg  t h a t c u r r e n t r e t u r n i n g through sheath 75%. is  earth sheath  T h i s agrees  s u b s t a t i o n s who  claim  i n s i d e the s u b s t a t i o n i s l e s s than  Based upon t h i s c r i t e r i a , a nominal e a r t h r e s i s t i v i t y  of 0.3  x 10 ^ftm  chosen. A f t e r choosing  a nominal v a l u e  current return c h a r a c t e r i s t i c and  increases.  earth.  for earth r e s i s t i v i t y ,  i s then e v a l u a t e d  the  as a f u n c t i o n of  depth, as shown i n Figure-3.6. F l u c t u a t i o n s i n o v e r a l l sheath  r e s u l t s a r e shown i n F i g u r e 3.6. even found t o be  At about 1 K Hz,  l a r g e r than the core c u r r e n t .  by the phasor diagram as shown i n F i g u r e 3.7.  the sheath  T h i s can be  sheath frequencies current  current i s  explained  In F i g u r e 3.7,only m u l t i -  c a b l e systems w i t h c u r r e n t i n c e n t r e core are shown, but  mutli-cable  system w i t h c u r r e n t s i n a l l 3 c o r e s would  Kralso  results.  t h a t a l l cores are decoupled  from one  The  present  study  a g a i n confirmed  another above 2 k Hz  reduced e a r t h r e s i s t i v i t y It sheaths and  identical  even f o r the adverse case of  i n s i d e the  significantly  substation.  should be n o t e d t h a t the mutual impedance between c o r e s , between between c o r r e s p o n d i n g  equal by Wedepohl and The v a l i d i t y  give  Ametani. The  cores and  sheaths a r e a l l assumed to  s h i e l d i n g e f f e c t o f the sheath  be  i s neglected.  of t h i s assumption i n c a b l e parameter computations c o u l d be  t o p i c of f u r t h e r r e s e a r c h . I t i s of l i t t l e  concern f o r the purpose of  t h e s i s . At the h i g h f r e q u e n c i e s encountered i n l i g h t n i n g c o r e c u r r e n t always r e t u r n completely the magnetic f i e l d  through  the  this  surge s t u d i e s ,  the sheath.  becomes zero o u t s i d e the sheath anyhow.  In that  case,  log 6  7  8  9  earth resistivity.*;  10  20  40  30  (10" ft-m) 6  F i g u r e 3.5:  R a t i o of c o r e , c u r r e n t r e t u r n through sheath a t 60 Hz f o r d i f f e r e n t e a r t h r e s i s t i v i t i e s .  77  z  o  sheath-mutual  sheath 120  sheath  core  earth  100  80  s i n g l e - c a b l e system t h r e e - c a b l e system i „/i  60  s2 (§) : depth or h e i g h t =  c2 .254  m  (E) : depth or h e i g h t = 1 m 40  © :  depth or h e i g h t = 6 m  p = 3uft-m  20  log  Figure  i  i  1  1  10  100  lk  10k  3.6:  frequency •  100k  R a t i o of c o r e c u r r e n t r e t u r n through own sheath f o r s i n g l e - and t h r e e - c a b l e system a t reduced e a r t h r e s i s t i v i t y o f 3 yft-m i n s i d e s u b s t a t i o n .  (Hz)  78  60 Hz  .1 Hz ->i  i  core  core  :". sheath  """so i i  more c o r e c u r r e n t r e t u r n through sheath than s o i l  more c o r e c u r r e n t r e t u r n through s o i l than s h e a t h  10k Hz  l k Hz  - —  ->  ->-  i  i  — 1  core —  core ,  '  sheath -y  Ii 1  J> i sheath  1  core  •+  """sheath " c c o r e 1  •"•core  """sheath  F i g u r e 3.7:  +  """soil  Phasor diagram o f c u r r e n t r e t u r n through sheath and e a r t h .  79 7  •  Formation of shunt admittance m a t r i x f o r SFfi c a b l e s For  a u s u a l 3-phase s i n g l e  core underground  c a b l e system, one  can  b u i l d a 6 x 6 shunt admittance m a t r i x [Y ] t o d e s c r i b e the c a b l e system  L  dx  [Y]  J  r d~  as  [V]  di  l  y  dx si • dx d  c dx  - l y  " l y  i  y  l  0  2  + y  2  0  y  0  0  0  0  0  0  0  0  0  0  l  I  cl  si  c2  i.e.  (3.60) d l  s2 dx  di _ c3 dx di  . s3 dx  0  0  0  0  0  0  0  0  0  0  - l y  y  l  +  y  2  0  0  l  - l  y  - l y  V  c3  y  y  l  + y  s2  2  s3  N o t i c e t h a t the o f f - d i a g o n a l submatrix of [Y] are a l l zero to the f a c t t h a t the grounded s h i e l d between c a b l e s . the  due  sheaths i n between a c t s as e l e c t r o s t a t i c  For the SF^ c a b l e system as shown i n F i g u r e 3.1,  d i a g o n a l submatrix elements  are i  y..  =  '1  J  iwc,  =  1  J  io)  2TT£  o  r  Jin  y  2  =  ^ 2 w c  =  2 7 r e 0  3  *  Jin where  y^ i s admittance due  to i n t e r n a l SF^ gas i n s u l a t i o n ,  i s admittance due to e x t e r n a l sheath (N.B. ',Th  e  external insulation  and  insulation.  i s n o n - e x i s t e n t f o r the SFg c a b l e . )  80 As has been confirmed by the author  i n p r e v i o u s f i n d i n g f o r sheath  c u r r e n t r e t u r n c h a r a c t e r i s t i c s , the core s h o u l d be r e p r e s e n t e d as phase.  Then, the admittance  reduced  to d  i  C  dx  =  e q u a t i o n shown i n E q u a t i o n  J  y  'l  C  r  8.  (3.60) s h o u l d be  V  In  y  single  self  — 2  * c  (3.62)  V  C o n f i r m a t i o n of n u m e r i c a l accuracy f o r c a b l e parameter c a l c u l a t i o n and c u r r e n t r e t u r n r a t i o s The n u m e r i c a l a c c u r a c y of the  computation  the c a b l e parameters o b t a i n e d by the developed  was  confirmed when  c a b l e c o n s t a n t s program,  32 and by the BPA  c a b l e c o n s t a n t program  three s i g n i f i c a n t  agree  c o n s i s t e n t l y to more than  figures. 38  Then, a 500  kV submarine c a b l e  was  chosen as another t e s t  In t h i s case, the c a b l e parameters f o r the submarine c a b l e was culated.  The  amount of core c u r r e n t r e t u r n i n g through  and the sea was  o b t a i n e d by t a k i n g i n t o account  grounded sheath and  the grounded armour.  c o r e c u r r e n t r e t u r n i n g through were o b t a i n e d as 14%, than two 9.  87.8%  and  The  the  r a t i o s of magnitudes of the sea at 60  Hz  These agreed t o more  48 findings.  S i n g l e phase r e p r e s e n t a t i o n parameters f o r m u l t i - p h a s e S i n c e a l l phases of the SF^ c a b l e s are decoupled  s i n g l e phase a b l e r e p r e s e n t a t i o n f o r s t u d y i n g c  cal-  of z e r o p o t e n t i a l s on  respectively.  f i g u r e s to the r e s u l t s o f o t h e r  first  the sheath, armour  the sheath ,the armour, and 5.6%  example.  SF6  cables  from one  another,  l i g h t n i n g o v e r v o l t a g e wave  81  propagation  i n SF, c a b l e i s recommended. "  can be c a l c u l a t e d  d  (3.1)  element  cl Z i + Z i + Z ( i „ + i „) + Z . , ( i . + i J — == s l l cl sl2 s i ml2 c2 s2 ml3 c3 s3 dx  0 = Z  i  s 2 1  c  l  + Z  Subtracting Equation  (  =  (  z  =  " s21>  sii  " W  Z  self  i  s 2 2  s  l  + Z  (3.64) from  sll  Z  dx  S i  +  ( Z  l  2  (i  c  + i ^ ) + Z  2  (3.63), we  sl2  - TT> cl  1  m  0  * s22 Z  )  "Wrr"  m  l  3  (i  c  3  from  (3.63)  +  {  ^  y  get  S i  Si>  < - > 3  65  cl  cl  where the sheath to core c u r r e n t r a t i o equations  self  as  V  and  (y  (3.61) and  impedance m a t r i x element can be c a l c u l a t e d  1  or  s e l f admittance  from the simple formula as shown i n e q u a t i o n s  (3.62), whereas the s e l f equation  The  (3.57) to  can be o b t a i n e d from e q u a t i o n  (3.59).  Because a l l core c u r r e n t r e t u r n s through f r e q u e n c i e s , one has  the sheath at h i g h  i n such c o n d i t i o n  Si i Substituting this  (3.49)  , = -1,' and  Z s l 2 = Z s22 .  cl  into equation  (3.65) or (3.37), one  . obtains  dV (  dx A f t e r h a v i n g o b t a i n e d the s e l f cable  as shown i n F i g u r e 3...1, one  Z  sll  " s22> Z  S  3  '  6  6  )  s e r i e s impedance f o r s i n g l e phase  can then r e p r e s e n t the c a b l e by the  surge impedance Z  (  , and the wave p r o p a g a t i o n v e l o c i t y v, as g i v e n by  surge  )  82  z  surge  /!self /  y  ^  =  fl  (  3  >  6  ?  )  self  v = oi / — = 300 m/ys / self • s e l f  (3.68)  y  One s h o u l d r e a l i z e t h a t the m e t a l l i c  sheath o f the SF^ c a b l e always  form a very good e a r t h r e t u r n p a t h to the c a b l e c o r e . i s n e g l i g i b l e compared t p t h e can be taken to be l o s s l e s s . go-return c i r c u i t simple f o r m u l a ^  The s e r i e s r e s i s t a n c e  r e a c t a n c e (See Fig.3,8),Thus, I t should a l s o be noted  the SF^ c a b l e  t h a t f o r such a simple  f o r a c o a x i a l c a b l e , the i n d u c t a n c e can be g i v e n by the as y L = •—- in - • 2TT =  r —  (3.69) r  0.205  2  V /m H  C o n s i s t e n t r e s u l t s f o r the i n d u c t a n c e are o b t a i n e d by e q u a t i o n and  (3.69) f o r f r e q u e n c i e s above 10 Hz.  equation  (3.65)  Thus, the simple formula i s i n  (3.69) i s recommended f o r i n d u c t a n c e c a l c u l a t i o n of SF^ c a b l e i n  the study of surge p r o p a g a t i o n c h a r a c t e r i s t i c s .  The surge impedance  and  7  the wave p r o p a g a t i o n v e l o c i t y can be then o b t a i n e d as  surge  =  =  /1  A ° y-  C  £  n  2TT  ° e  ..  1 , 2  o  4TT  I  r  „  . _1_  3  2  2TTE  . o  £  n  I  3  r  2  3 r„ 2  r 3  = 60 £n — 2 r  where r ^ and r  2  a r e r a d i u s of sheath and c o r e r e s p e c t i v e l y . 10  (3.70)  Figure  3.8:  S e l f s e r i e s i n d u c t a n c e , r e s i s t a n c e and r e s i s t a n c e to r e a c t a n c e r a t i o f o r SF, c a b l e .  84  and  v=  /f^ = •'  T  / — /  r  Jin—  y  2TTE  r„ r  (3.70)  r„  2  = 300 / y s m  y £ o o  v e l o c i t y of l i g h t  10.  Wave p r o p a g a t i o n  i n SF6 c a b l e s  Wave p r o p a g a t i o n  c h a r a c t e r i s t i c s i n s i n g l e core SF^ c a b l e can now  be modelled by t h e surge impedance o f 61.4 and wave p r o p a g a t i o n  i n vacuum  velocity,  ft,(typically  ( t y p i c a l l y 300 / y s ) . m  about 60 to 75 ft),  A numerical  simula-  t i o n o f o v e r v o l t a g e wave-shape .in t h e r e c e i v i n g end o f a SF^ c a b l e  joining  to a overhead t r a n s m i s s i o n l i n e i s s i m u l a t e d . The  r e s u l t i n g v o l t a g e i n t h e o p e n - c i r c u i t e d SF^ c a b l e r e c e i v i n g end  r i s e s i n a s t a i r c a s e f a s h i o n o f d i m i n i s h i n g amplitude, (See F i g u r e 3.9).  t o a v a l u e o f 2 p.u.  T h i s can be e x p l a i n e d by u s i n g t h e r e f l e c t i o n  refraction coefficient  (C.) and  (C ) o f t h e system a t t h e l i n e - c a b l e j u n c t i o n and is.  39  the o p e n - c i r c u i t e d c a b l e end r e s p e c t i v e l y .  For the l i n e - c a b l e junction  at A, one has Z  2  = 312, Z  1  = 60ft  (Wave i n c i d e n t from c a b l e )  (Wave i n c i d e n t from l i n e ) Z = 60, Z = 312 ft 2  Z c  ^  =  z  - Z + 7^  =  312 '+" 60  =  (3- ) 7 ±  85  Figure'3.9:  O v e r v o l t a g e waveshapes a t b o t h ends o f SF^ c a b l e j o i n i n g from overhead t r a n s m i s s i o n l i n e .  86  For t h e open  end of t h e c a b l e , we have 1 2 ,o C R  and  2 x °° + 60  (Wave i n c i d e n t  Thus, the d i s c r e t e r i s e  °° - 60 = 1 °° + 60  2 from c a b l e , Z  = 60, Z  i n v o l t a g e wave shape can be expressed as  v = 2 x .32 (1 + C . +  C.  2  + . . .)  where each step a d d i t i o n a c c u r s a t d i s c r e t e time travel  = »)  intervals  of 2  times.  On the o t h e r hand, t h i s  overall  r i s e i n o v e r v o l t a g e s wave shape  a l s o agrees w i t h t h e g e n e r a l e x p o n e n t i a l r i s e wave shape i n c h a r g i n g o f a capacitor. The o v e r a l l  T h i s i s due t o t h e i n h e r e n t l a r g e s e l f  rise  i n wave shape can  : be  sketched  capacitance of c a b l e s . by m o d e l l i n g t h e  SFg c a b l e as a lumped c a p a c i t o r e q u i v a l e n t t o the t o t a l c a p a c i t a n c e f o r the l e n g t h o f c a b l e , and i g n o r i n g the surge F i g u r e 3.9).  impedance o f the c a b l e (See  87 CHAPTER 4:  1.  CORONA ATTENUATION AND DISTORTION CHARACTERISTICS OF LIGHTNING OVERVOLTAGE IN OVERHEAD TRANSMISSION LINES.  Introduction As the l i g h t n i n g v o l t a g e wave t r a v e l s down the overhead t r a n s m i s s i o n  l i n e , a high e l e c t r i c f i e l d  i s produced  on the l i n e conductor s u r f a c e .  When kV  the e l e c t r i c  field  i n t e n s i t y exceeds  the breakdown s t r e n g t h of a i r (^30  i o n i z a t i o n of s u r r o u n d i n g a i r m o l e c u l e s takes p l a c e . d i s s i p a t e the unwanted surge energy  /em),  T h i s phenomenon w i l l  away from the system and thus  reduces  the magnitude and i n i t i a l r a t e of r i s e o? t h e l i g h t n i n g o v e r v o l t a g e . In t r a n s i e n t l i g h t n i n g o v e r v o l t a g e s t u d i e s , s e v e r a l n u m e r i c a l methods 44 have been employed to account of  f o r corona e f f e c t s .  Brown  a p p l i e d the  corona r a d i u s to account f o r the corona envelope produced  surface.  on the  concept  conductor  The c o r o n a t e d l i n e c a p a c i t a n c e s at h i g h e r v o l t a g e s are a l s o o b t a i n e d '•  12 by e x t r a p o l a t i o n . D a r v e n i z a  a l s o used lower wave p r o p a g a t i o n v e l o c i t i e s  h i g h e r v o l t a g e s and d i f f e r e n t corona c o r r e c t i o n f a c t o r f o r d i f f e r e n t tor  configuration.  conduc-  However, both methods are not s t r a i g h t f o r w a r d and are 43  not t o t a l l y s u c c e s s f u l i n d u p l i c a t i n g f i e l d  rest results.  Umoto and  a l s o t r a n s f o r m e d the t r a n s m i s s i o n l i n e e q u a t i o n f o r coronated l i n e s difference algebraic equations. efficient  enough.  However, t h i s n u m e r i c a l approach  Thus, an e f f i c i e n t  Hara into  i s not  and a c c u r a t e n u m e r i c a l model f o r corona  must be developed t o p r e d i c t the corona a t t e n u a t i o n and d i s t o r t i o n c h a r a c t e r i s t i c s on l i g h t n i n g o v e r v o l t a g e p r o p a g a t i o n s i n overhead l i n e s . 2. P h y s i c a l p r o p e r t i e s o f corona a t t e n u a t i o n and d i s t o r t i o n c h a r a c t e r i s t i c s The p h y s i c a l a s p e c t s and laws governing the b e h a v i o u r of corona charge have been i n v e s t i g a t e d s i n c e the b e g i n n i n g o f t h i s c e n t u r y .  dis-  However,  most o f the i n v e s t i g a t i o n s and a p p l i c a t i o n s have been l i m i t e d t o power frequency steady s t a t e or a t most t o s w i t c h i n g t r a n s i e n t c o n d i t i o n s .  From  o  88 the p u b l i s h e d f i e l d measurements f o r l i g h t n i n g surges, t h a t the a t t e n u a t i o n r e s u l t i n g from corona  i t can be  e f f e c t s i s much l a r g e r than t h a t  r e s u l t i n g from t r a n s m i s s i o n l i n e s e r i e s r e s i s t a n c e l o s s e s . c h a r a c t e r i s t i c s o f t h e corona  a)  discharge  observed  The n o n - l i n e a r  can be c o n s i d e r e d as (see F i g u r e 4 . 1 ) :  Corona a t t e n u a t i o n l o s s - From the q u a d r a t i c law o f corona  loss pro-  41 posed by Peek  , the l o s s  (v i ^ ) p e r u n i t l e n g t h i s p r o p o r t i o n a l t o t h e  square of the v o l t a g e above the c r i t i c a l  where  = a  k  r  /m  = Corona l o s s constant determined  experimentally  a t t e n u a t i o n l o s s can be modelled w i t h a r e s i s t i v e c u r r e n t l o s s i ^  through the corona  b)  i.e.  , h = r a d i u s and h e i g h t o f conductor r e s p e c t i v e l y a  T h i s corona  • /-^r x 10  corona v o l t a g e v  Increase  r e s i s t i v e branch to ground as  i n shunt c a p a c i t a n c e - the r e t a r d a t i o n o f the wave f r o n t by 42  corona  can be e x p l a i n e d by an i n c r e a s e i n shunt c a p a c i t a n c e .  Skilling  43 and Umoto  suggested  t h a t the i n c r e a s e i n shunt c a p a c i t a n c e i s  p r o p o r t i o n a l t o t h e v o l t a g e above the c r i t i c a l V = 2k (1 where corona k = a c xV 10 c c v 2h C  a  c  = corona  voltage V  c  q  , i.e.  (4.3)  h  l o s s constant^determined  experimentally  89  V - V  - v  R  1  Corona  + KR  shunt c a p a c i t a n c e V  C  c  transmission line  c  corona  = 2k  c  (1  /  G  „ V.  rrfn  F i g u r e 4.1:  = n--~f k  corona  rmr  N o n l i n e a r corona l o s s e s model.  R  V  90  T h i s i n c r e a s e i n c a p a c i t a n c e can be modelled branch  to ground w i t h the  by : a c a p a c i t a n c e  c a p a c t i v e c u r r e n t l o s s i„  Corona d i s c h a r g e o n l y occur i f the v o l t a g e i s g r e a t e r o r e q u a l to the c r i t i c a l  corona v o l t a g e , and i f the v o l t a g e i n c r e a s e w i t h time, i . e .  v £ v  co  j 3v , and — > o. 3t  T h i s i s due t o t h e f a c t t h a t , when the v o l t a g e b e g i n s  to decrease, the  space charge c o n s i s t i n g o f heavy i o n s i n the i o n i z a t i o n r e g i o n remains practically time.  constant  i n magnitude and p o s i t i o n d u r i n g a s h o r t p e r i o d o f  T h i s slow d i f f u s i o n o f i o n s r e s u l t s i n l i t t l e energy l o s s i n t h e  case o f d e c r e a s i n g v o l t a g e c o n d i t i o n s even when v > V  3.  Transmission The  corona  l i n e equations.  l i n e equations  f o r coronated  C  Q  .  lines.  phenomena can now be d e s c r i b e d by t h e m o d i f i e d With the i n t r o d u c t i o n o f d i g i t a l  phenomena can be s t u d i e d  computers,  these  a c c u r a t e l y by s o l v i n g t h e e q u a t i o n s d e s c r i b i n g  the e l e c t r o m a g n e t i c wave p r o p a g a t i o n s  t a k i n g corona  i n t o account as  follows:  = £ + ( i - ^ S + v -'^ -' 1  e x t r a shunt capacitance due t o corona  E x t r a shunt conductance due t o corona  2  ->  < 4 6  43  45  Umoto and Inoue method.  s o l v e d the above e q u a t i o n s by the d i f f e r e n c e  The l i n e e q u a t i o n s (4.5) and  (4.6) a r e transformed i n t o  e q u a t i o n s of s m a l l increments of d i s t a n c e , Ax, t h i s method i s not e f f i c i e n t  l i n e e q u a t i o n s (4.5) and  s o l v e d by the compensation are f i r s t  when  r  S o l u t i o n of l i n e e q u a t i o n by compensation The  However,  to implement i n t o the d i g i t a l computer as  the method r e q u i r e s Ax t o be as s m a l l as 7 m 4.  and time, At.  algebraic  using  At = .01 y s .  method w i t h t r a p e z o i d a l  rules  (4.6) w i t h corona l o s s e s can be  method.  In t h i s method, the l i n e  s o l v e d w i t h o u t the e x t r a corona terms.  equations  The Bergeron's method  u s i n g t r a v e l l i n g wave t e c h n i q u e t o g e t h e r w i t h modal a h a l y s i s ( S e e Chapter is  applied.  shunt branches  2)  Then, the corona l o s s e s can be t r e a t e d as n o n - l i n e a r connected t o ground,. .< The  trapezoidal rule  can then  a p p l i e d to o b t a i n the t o t a l c u r r e n t l o s s of t h e corona phenomena. By a p p l y i n g the t r a p e z o i d a l r u l e of l i n e a r i n t e r p o l a t i o n to the corona r e s i s t i v e branch to ground, we have j as shown i n F i g u r e  ±  where  - -  d =  as  e  v  t+A  +  v  t + At  = t  i  W  +  (  v  t " i  4.2,  V  — — v t  (4.7)  d = s l o p e of graph at time t .  A l s o , d can be o b t a i n e d by c o n s i d e r i n g the e q u a t i o n  (4.2)  92  Current  (  i(v)  Voltage v  F i g u r e 4.2:  L i n e a r i n t e r p o l a t i o n f o r r e s i s t a n c e corona b r a n c h .  8v/9t=v  v o l t a g e v or current i  F i g u r e 4.3:  L i n e a r i n t e r p o l a t i o n f o r c a p a c i t i v e corona b r a n c h .  93  t  R  ( v t  k^ v  - v  )' CO  v.  + k„ — R  t  co  - 2k„ v ~"R co  v  di d =  or  v 2  dv  k  K.  R  (4.8)  v 2  Thus, we e v e n t u a l l y have 1 V  t . + At "  .  \  where  v  and  ^R  +  (  i  t  a"  =  + At  ' t + At  = v  o  , ,v  \  d  - 1 i v d °  fc  +  - v i , d t  V  (4.9)  (4.10)  o  ( known from past h i s t o r y a t time t )  (known from p a s t h i s t o r y a t time t )  S i m i l a r l y , s i n c e the corona c a p a c i t i v e branch  current loss i s  g i v e n by 2  x =  or  9v 3t  k  v  c , (v - v  v i 2k (v - v c  , a9v ) — co 3t  CO  )  f(v, i)  A p p l y i n g l i n e a r i n t e r p o l a t i o n of the 2 v a r i a b l e s ( i . e . from term o f T a y l o r s ' s e r i e s ) , we have as shown i n F i g u r e  = f(v, i)  f(v, i) t+At  + 3v  (v  t+At  V  \ t>  first  4.3  +  , 3f ' 31  ^t+At  " t ±  )  (4.11)  94  3f 3v  i  (v  2k  t  v  c  C O  - X ,  2  and  3f 3i  ) - v  k  V  c  <v  t  C O  - v  c o  (4.12)  )2  2kj;v - v ) * co  t  2  (4.13)  H\  -  \o>  Thus, we o b t a i n 3v 3t  f(v, i ) t+At  v t+At  t+At "At  — v t  v. 2k(v - v ) ^ t co  v  t+At  / t /  Re-arranging e q u a t i o n (4.14) w i l l g i v e the l i n e a r i z e d e q u a t i o n as  V  where  t+At  R  =  R  c S+At  + V  (4.15)  l  * v At t (known from p a s t 2k (v - v ) , . „ . c t co hxstory)  = 1 +  v i * . . co t At 2k (v - v )2 C t C O  .  and  (  1 +  co K't A« At 2k (v - v )2 c t co v  V  t  +  . V  X  V  At co t t 2k (v - v )2> c t co  from past hxstory)  Combining equations  (4.10) and  (4.15) f o r the v o l t a g e s and c u r r e n t s  i n both corona r e s i s t i v e and c a p a c i t i v e branches by t a k i n g i n t o  v  and  we  = v_. = v R  i = i  +  c  i„ R  can o b t a i n  (^  of  c  account  + ^)  or  v-  (i  R' R  and  Having  i )  +  R  (^  +  +  ^ )  v = R ' i + k'  R  where  c  k* =  c  *R  c  (4.17)  *R k  +  R  V  l  +  R  c  V  o  R  the corona l o s s branches r e p r e s e n t e d by a l i n e a r model as  described i n equation  (4.17), the compensation method can then be a p p l i e d  to s o l v e the t r a n s m i s s i o n l i n e e q u a t i o n s i n c l u d i n g corona  losses.  In the compensation method, the t r a n s m i s s i o n l i n e i s f i r s t to a T h e v i n i n e q u i v a l e n t (See F i g u r e 4.4)  vv=  where  V  Q  and  reduced  i s d e s c r i b e d by  + A ±,  (4.18)  2  i s a n e g a t i v e number.  Then, t h i s e q u a t i o n i s s o l v e d s i m u l t a n e o u s l y w i t h the e q u a t i o n f o r corona l o s s , as i n e q u a t i o n  (4.17).  Thus, the  corona v o l t a g e and d i s c h a r g e c u r r e n t can be o b t a i n e d as  linearized resulting  A  m (m i s ground)  A : T h e v i n i n e q u i v a l e n t network f o r t r a n s m i s s i o n without corona l o s s e s  B : Nonlinear  line  corona l o s s e s model v = R' i + k'  V o l t a g e v.  v  corona  «• v  •»vkm  =v +A i o  2  Current i  F i g u r e 4.4:  i  km  corona  Compensation method f o r n o n - l i n e a r corona model  1  - R' - A.  R'v  A  and  5.  corona  v  corona  • R  o  1  (4.19)  - A-k* 2 - A.  (4.20)  I n f l u e n c e on corona by a d j a c e n t sub-conductors i n the same bundle E x t r a - h i g h v o l t a g e phase conductors a r e designed t o c o n s i s t o f  s e v e r a l sub-conductors bundled t o g e t h e r i n o r d e r t o reduce corona l o s s e s . The e l e c t r i c the  f i e l d on a sub-conductor  s u r f a c e i s a f f e c t e d a p p r e c i a b l y by  a d j a c e n t sub-conductors i n the same b u n d l e s .  consequently  The corona phenomenon i s  influenced.  The e l e c t r i c conductor i t s e l f  field  on t h e sub-conductor s u r f a c e due t o the sub-  i s g i v e n by  46  cv  2 IT where  , Q-= c h a r g e / l e n g t h r  2 IT  max  e o  r  e o  r  (4.21)  c = effective capacitance/length v = v o l t a g e o f conductor r = r a d i u s o f sub-conductor  However, f o r a bundled conductor w i t h 4 i n d i v i d u a l the  maximum e l e c t r i c  field  i s g i v e n by^(See F i g u r e 4.5) Q  max  sub-conductors,  1  2TT e  -21  •- -2-TT  e r o  ,- 1  +  s72  + 2  -•  s i n 45 ) u  (4.22)  98  \ a x - -2^tT  (  1  „, where Q = F i g u r e 4.5:  +  vfs C  }  effective ^  V  C r i t i c a l v o l t a g e c a l c u l a t i o n by e v a l u a t i o n o f maximum e l e c t r i c f i e l d on a 4-conductor bundle.  A f t e r the maximum e l e c t r i c obtained,  field  on the conductor s u r f a c e i s  the c r i t i c a l v o l t a g e f o r corona d i s c h a r g e  can be computed  by  6 equating  the maximum e l e c t r i c f i e l d  t o 30 kV/cm or 3 x 10  e l e c t r i c breakdown s t r e n g t h i n a i r . conductor has be  558  6.  been found to be 277  v/m,  A t y p i c a l c r i t i c a l voltage f o r a single  kV,  and  t h a t f o r a 4-conductor bundle  I n f l u e n c e on corona by adjacent  v o l t a g e s are always induced electric field  phase conductors  i n the a d j a c e n t  coupled  conductors.  on the conductor s u r f a c e i s a f f e c t e d .  However, due  between phase conductors are u s u a l l y l a r g e compared w i t h conductors.  But  to  the  r a d i u s of  individual  overvoltages  t h i s change i n c r i t i c a l v o l t a g e produces n e g l i g i b l e  e f f e c t s on the o v e r a l l corona a t t e n u a t i o n and o v e r v o l t a g e wave (See F i g u r e  distortion characteristics  on  4.6).  Optimal lumping l o c a t i o n s and number of corona branch l e g s The  equations  dbmbensation method  w i t h corona phenomenon i s now  s o l v e d by  w i t h the corona l o s s l e g s lumped a t a few  along t h e transmission•-• l i n e .  However,  number of lumped elements  to be determined.  At f i r s t , 20  has  the places  the o p t i m a l l o c a t i o n s and  corona l o s s branches 70 m apart  were lumped between f i v e t r a n s m i s s i o n 350  another,  separating  T h i s e f f e c t u s u a l l y change the o v e r a l l c r i t i c a l  l e s s than 10%.  to one  Thus the maximum  d e s i g n of t r a n s m i s s i o n l i n e s f o r e x t r a h i g h v o l t a g e l e v e l s ,  7.  to  kV.  S i n c e the conductors i n each phase are m u t u a l l y  by  . the  towers.  m between the corona l o s s branches was  from one  another  Then, a s e p a r a t i o n of  used.  This increase i n  t i o n i n c r e a s e d the d e v i a t i o n of the p r e d i c t e d wave shape from f i e l d ments' a p p r e c i a b l y  (See F i g u r e 4.7),  optimal  from about 5 t o 10%.  separameasure-  T h i s suggests that  f i e l d measurements v i 303-kV ( i n c l u d e e f f e c t of. c r i xca adjacent phase c o n d u c t o r s ) ) v . . 277 kV ( n e g l e c t e f f e c t of crxtical adjacent phase c o n d u c t o r s ) ± t ± c a  F i g u r e 4.6:  Effect  =  o f adjacent phase conductors on corona  losses.  F i g u r e 4.7:  E f f e c t of lumping d i s t a n c e s on corona.  102 the o p t i m a l s e p a r a t i o n s h o u l d be about The  f i e l d measurement f o r a 4-conductor  In u s i n g the corona l o s s c o n s t a n t s  a  c  a-  s l i g h t l y higher constants  70 m.  ( f o r case of 1-conductor  = 10 x 1 0  ( f o r case o f 4-conductors  c  a  bundle)  = 30  6  o v e r v o l t a g e s were o b t a i n e d .  a  bundled was then s i m u l a t e d .  Then, a new s e t o f corona  bundle)  = 30  = 20 x 1 0  6  was used to g i v e r e s u l t s c o n s i s t e n t w i t h those from f i e l d measurements (See F i g u r e 4.8). F i n a l l y , the n e g a t i v e impulse o v e r v o l t a g e was a l s o s i m u l a t e d f o r the 4-conductor  bundle case.  The corona l o s s i n t h i s case was found t o be  much l e s s than the p o s i t i v e Impulse c a s e .  The corona l o s s c o n s t a n t s were  determined t o be  a  c  a  = 15  = 10 x 1 0  6  With these s e t s of corona c o n s t a n t s , the f i e l d was a g a i n r e p l i c a t e d c l o s e l y  8.  (See F i g u r e 4.9).  O v e r a l l n u m e r i c a l m o d e l l i n g f o r corona The  field  t e s t measurement  effects  t e s t r e s u l t s o f corona a t t e n u a t i o n and d i s t o r t i o n  c h a r a c t e r i s t i c s on a 500 kV t e s t l i n e were r e p l i c a t e d by the method  Overvoltages(kV)  F i g u r e 4.8:  P o s i t i v e impulse on 4- conductor  bundle."  Overvoltages(kV) 2000 FDUR-CONDUCTOR BUNDLE  1  2  3  4  field  5  measurements  a =15, a = 1 0 x l 0 c G r a =15, a =5x10 c G  F i g u r e 4.9:  Negative impulse on 4 T conductor  6  bundle.  6  105  developed e a r l i e r .  T h i s method examined corona c h a r a c t e r i s t i c s  s i n g l e and bundled conductor l i n e s .  From the  performed  study, one  conclude t h a t the e f f e c t s of b u n d l i n g of conductors i s e f f i c i e n t ing c r i t i c a l  corona v o l t a g e .  Furthermore,  conductors i s n e g l i g i b l e on corona e f f e c t s .  i s determined  can  in increas-  i n f l u e n c e o f a d j a c e n t phase Thus, i t i s concluded t h a t  s i n g l e phase l i n e r e p r e s e n t a t i o n i s s u f f i c i e n t it  i n both  f o r corona s t u d i e s .  Finally,  t h a t s e p a r a t i o n between the corona l o s s l e g can be lumped  at 70 m w i t h o u t s a c r i f i c i n g a l o s s of a c c u r a c y on the p r e d i c t e d c o r o n a t e d waveform. A. r e d u c t i o n i n d i s t a n c e between corona l e g s w i l l not improve t h e a c c u r a c y of the s i m u l a t e d r e s u l t s . It  should be noted t h a t l i g h t n i n g  one conductor  a t one time; thus corona phenomena have o n l y been i n c l u d e d  f o r one conductor simultaneously.  s t r o k e s w i l l r a r e l y h i t more than  i n t h i s t h e s i s , r a t h e r than f o r a l l t h r e e phases  106 CHAPTER 5:  CONCLUSIONS  The a t t e n t u a t i o n and d i s t o r t i o n of l i g h t n i n g o v e r v o l t a g e waves on m u l t i - p h a s e  t r a n s m i s s i o n l i n e s and m u l t i - p h a s e  i n compressed SF^ g a s - i n s u l a t e d s u b s t a t i o n s was of l i g h t n i n g o v e r v o l t a g e s on overhead  s i n g l e c o r e SF^ c a b l e s  studied.  l i n e s were a l s o  Corona e f f e c t s  investigated.  A v a i l a b l e f i e l d t e s t r e s u l t s f o r corona e f f e c t s were d u p l i c a t e d t o w i t h i n 5%  accuracy. R e s u l t s o b t a i n e d w i t h the t e c h n i q u e s developed  by the author  21  useful for lightning insulation co-ordination studies  ,. 13,14,40.  ^  i t  related  J  xhe l i g h t n i n g surge wave f r o n t can be c a l c u l a t e d a t  studies  l o c a t i o n i n s i d e the s u b s t a t i o n , eg., former  ,  and o t h e r  are  terminal.  i n s i d e the SF^ bus o  any  or a t the t r a n s -  Based on the s t u d i e s d e s c r i b e d i n the t h e s i s the f o l l o w i n g  recommendations are made f o r f u t u r e i n s u l a t i o n c o - o r d i n a t i o n d e s i g n s t u d i e s : 1.  M u l t i - p h a s e untransposed  l i n e s can be r e p r e s e n t e d by s i n g l e - p h a s e  models u s i n g s e l f parameters c a l c u l a t e d a t a h i g h frequency approximately ignored.  1 M Hz  (See T a b l e 2.4).  g a t i o n over d i s t a n c e s l e s s than 2 2.  Corona e f f e c t s a r e important r i s e o f the incoming  of  S e r i e s r e s i s t a n c e should  Frequency dependent e f f e c t s are not important  line  be  f o r propa-  km.  i n r e d u c i n g t h e magnitude and r a t e o f  l i g h t n i n g o v e r v o l t a g e surge.  Efficient  t e c h n i q u e s u s i n g compensation methods a r e developed  solution  to s o l v e the non-  l i n e a r corona a t t e n u a t i o n and d i s t o r t i o n phenomenon. 3.  Multi-phase  SF^ s i n g l e c o r e c a b l e s can be r e p r e s e n t e d by s i n g l e phase  c a b l e models.  S e r i e s r e s i s t a n c e can be i g n o r e d .  Cable parameter can  o b t a i n e d w i t h the simple formula f o r a g o - r e t u r n c i r c u i t cable with s u f f i c i e n t  accuracy.  for a coaxial  be  107 APPENDIX A:  SKIN DEPTH ATTENUATION IN CONDUCTING MEDIUM WITH FINITE CONDUCTIVITY.  This section  shows  t h a t the core c u r r e n t r e t u r n c h a r a c t e r i s t i c s  through t h e sheath f o r the SF^ c a b l e c o u l d be o b t a i n e d by a d i f f e r e n t approach.  From the Maxwell's equations i n a c o n d u c t i n g medium, we  V x E = -jwu H  have^  (A.l)  V x H = jtoeE + aE = aE, f o r good conductors  (A.2)  where a i s c o n d u c t i v i t y o f medium. From e q u a t i o n s  ( A . l ) and (A.2), we c a n get 2  2  V x V x E = V ( V « E ) - V E  =-VE  ( f o r homogeneous medium)  = -joiyV x H = -jwyaE = where  -m E 2  m = /jtoya = ^  ^ • /u>ya  T h i s e q u a t i o n i s i d e n t i c a l t o the d i f f u s i o n e q u a t i o n w i t h solutions —  a  = E  = E ex o  = E -V"2~  m Z  ,  Z  e  o  = E eo  j Z / 6  • e"  (A.3)  • e-V~T j Z / 6  Z  (A.4) (A.5)  108 6 =  where  ~-.  =  -  1  Thus, the t a n g e n t i a l e l e c t r i c f i e l d density J equals  x  w i l l be a t t e n u a t e d  7= s k i n depth  E  o r the t a n g e n t i a l c u r r e n t  by ^ = -368 when the depth o f p e n e t r a t i o n Z  F o r aluminum, we have the s k i n depth 6 as  to the s k i n depth.  <5 = 7 = =  (A.6)  /irf (4ITX10-7) (3-8x10/) 8-1 ,—  Therefore, attenuated istic  f o r frequency  cm  above 1 kHz, the e l e c t r i c  and n e g l i g i b l e f l u x o u t s i d e the sheath.  field  Thus, s i n c e  i s essentially character-  f r e q u e n c i e s o f l i g h t n i n g s t r o k e s exceeds 1 kHz, t h e above r e s u l t s  i n d i c a t e t h a t each phase of the c a b l e i s decoupled from o t h e r phases as was shown p r e v i o u s l y i n Chapter 3. A f t e r the t a n g e n t i a l c u r r e n t d e n s i t y f o r one medium i s o b t a i n e d by e q u a t i o n s  (A.3) to (A.5), the t a n g e n t i a l c u r r e n t d e n s i t y f o r another  medium on t h e boundary t o t h e f i r s t medium can be o b t a i n e d by  E  lt  J  lt  = 2t E  °1 =  ~  J 2  2t  Thusm the t o t a l c u r r e n t f l o w i n g i n d i f f e r e n t components o f t h e c a b l e system can be o b t a i n e d by I = /JdA  109  BIBLIOGRAPHY  1.  W. D i e s e n d o r f , ' I n s u l a t i o n c o o r d i n a t i o n i n h i g h v o l t a g e e l e c t r i c power systems', B u t t e r w o r t h Cp. London 1974.  2.  M. Uman, 'Understanding P e n n s y l v a n i a , 1971.  3.  'Transmission and D i s t r i b u t i o n Reference Book', Westinghouse E l e c t r i c C o r p o r a t i o n , Pennsylvania,1964.  4.  F. 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